Method and apparatus for transmitting uplink signals using multi-antenna

ABSTRACT

A method and apparatus for allowing a UE to transmit uplink signals using a MIMO scheme are disclosed. In order to maintain good Peak power to Average Power Ratio (PAPR) or Cubic Metric (CM) properties when the UE transmits uplink signals using the MIMO scheme, the UE uses a precoding scheme based on a precoding matrix established in a manner that one layer is transmitted to each antenna in specific rank transmission.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of Korean Patent Application Nos.10-2008-0132994 and 10-2009-0073606, filed on Dec. 24, 2008 and Aug. 11,2009, respectively, which are hereby incorporated by reference as iffully set forth herein.

This application claims the benefit of U.S. Provisional Application Nos.61/087,990, 61/160,711, 61/169,726, 61/170,106 and 61/173,585, filed onAug. 11, 2008, Mar. 17, 2009, Apr. 16, 2009, Apr. 17, 2009 and Apr. 28,2009, respectively, which are hereby incorporated by reference as iffully set forth herein.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a wireless mobile communication system,and more particularly to a communication system based on a MultipleInput Multiple Output (MIMO) scheme.

2. Discussion of the Related Art

MIMO technology is an abbreviation for Multiple Input Multiple Outputtechnology. MIMO technology uses a plurality of transmission (Tx)antennas and a plurality of reception (Rx) antennas to improve theefficiency of transmission and reception (Tx/Rx) of data. In otherwords, MIMO technology allows a transmission end or reception end of awireless communication system to use multiple antennas (hereinafterreferred to as a multi-antenna), so that the capacity or performance canbe improved. For convenience of description, the term “MIMO” can also beconsidered to be a multi-antenna technology.

In more detail, MIMO technology is not dependent on a single antennapath to receive a single total message. Instead, the MIMO technologycollects a plurality of data fragments received via several antennas,merges the collected data fragments, and completes total data. As aresult, MIMO technology can increase a data transfer rate within apredetermined-sized cell region, or can increase system coverage whileguaranteeing a specific data transfer rate. Under this situation, MIMOtechnology can be widely applied to mobile communication terminals,repeaters, or the like. MIMO technology can extend the range of datacommunication, so that it can overcome the limited amount oftransmission (Tx) data of mobile communication systems.

FIG. 1 is a block diagram illustrating a general MIMO communicationsystem.

Referring to FIG. 1, the number of transmission (Tx) antennas in atransmitter is N_(T), and the number of reception (Rx) antennas in areceiver is N_(R). In this way, theoretical channel transmissioncapacity of the MIMO communication system when both the transmitter andthe receiver use a plurality of antennas is greater than that of anothercase in which only the transmitter or the receiver uses severalantennas. The theoretical channel transmission capacity of the MIMOcommunication system increases in proportion to the number of antennas.Therefore, data transfer rate and frequency efficiency are greatlyincreased. Provided that a maximum data transfer rate acquired when asingle antenna is used is set to R_(o), a data transfer rate acquiredwhen multiple antennas are used can theoretically increase by apredetermined amount that corresponds to the maximum data transfer rate(R_(o)) multiplied by a rate of increase R_(i). The rate of increase(R_(i)) can be represented by the following equation 1.

R _(i)=min(N _(T) ,N _(R))  [Equation 1]

For example, provided that a MIMO system uses four transmission (Tx)antennas and four reception (Rx) antennas, the MIMO system cantheoretically acquire a high data transfer rate which is four timeshigher than that of a single antenna system. After the above-mentionedtheoretical capacity increase of the MIMO system was demonstrated in themid-1990s, many developers began to conduct intensive research into avariety of technologies which can substantially increase a data transferrate using the theoretical capacity increase. Some of the abovetechnologies have been reflected in a variety of wireless communicationstandards, for example, a third-generation mobile communication or anext-generation wireless LAN, etc.

The above-mentioned MIMO technology can be classified into a spatialdiversity scheme (also called a Transmit Diversity scheme) and a spatialmultiplexing scheme. The spatial diversity scheme increases transmissionreliability using symbols passing various channel paths. The spatialmultiplexing scheme simultaneously transmits a plurality of data symbolsvia a plurality of transmission (Tx) antennas, so that it increases atransfer rate of data. In addition, the combination of the spatialdiversity scheme and the spatial multiplexing scheme has also beenrecently developed to properly acquire unique advantages of the twoschemes.

In association with the MIMO technology, a variety of MIMO-associatedtechnologies have been intensively researched by many companies ordevelopers, for example, research into an information theory associatedwith a MIMO communication capacity calculation under various channelenvironments or multiple access environments, research into radiofrequency (RF) channel measurement and modeling of the MIMO system, andresearch into a space-time signal processing technology for increasingtransmission reliability and data transfer rate.

In a 3^(rd) Generation Partnership Project Long Term Evolution (3GPPLTE) system, the above-mentioned MIMO scheme is applied to only downlinksignal transmission of the 3GPP LTE system. The MIMO technology may alsobe applied to uplink signal transmission. In this case, a transmitterstructure is changed to implement the MIMO technology, so that a Peakpower to Average Power Ratio (PAPR) or Cubic Metric (CM) characteristicsmay be deteriorated. Therefore, there is needed a new technology capableof effectively applying the MIMO scheme to uplink signal transmission.

SUMMARY OF THE INVENTION

Accordingly, the present invention is directed to a method and apparatusfor transmitting uplink signals via multiple antennas that substantiallyobviate one or more problems due to limitations and disadvantages of therelated art.

An object of the present invention is to provide a technology foreffectively carrying out uplink signal transmission according to a MIMOscheme.

Additional advantages, objects, and features of the invention will beset forth in part in the description which follows and in part willbecome apparent to those having ordinary skill in the art uponexamination of the following or may be learned from practice of theinvention. The objectives and other advantages of the invention may berealized and attained by the structure particularly pointed out in thewritten description and claims hereof as well as the appended drawings.

To achieve these objects and other advantages and in accordance with thepurpose of the invention, as embodied and broadly described herein, amethod for enabling a user equipment (UE) to transmit uplink signals viamultiple antennas includes mapping the uplink signals to a predeterminednumber of layers, performing Discrete Fourier Transform (DFT) spreadingupon each of the predetermined number of layer signals, precoding theDFT-spread layer signals by selecting a specific precoding matrixestablished in a manner that one layer signal is transmitted to each ofthe multiple antennas from among a prestored codebook, and performing apredetermined process for constructing a Single Carrier-FrequencyDivision Multiple Access (SC-FDMA) symbol upon the precoded signals, andtransmitting the processed signals to a base station (BS) via themultiple antennas.

The specific precoding matrix may be a precoding matrix established in amanner that the multiple antennas have uniform transmission powertherebetween. The specific precoding matrix may be a precoding matrixestablished in a manner that the predetermined number of layers haveuniform transmission power therebetween

The codebook may include a first type precoding matrix, wherein thefirst type precoding matrix may be configured in a form of

$\begin{bmatrix}1 & 0 \\X & 0 \\0 & 1 \\0 & Y\end{bmatrix},$

as a Rank 2 precoding matrix utilized when the number of the multipleantennas is 4 and a rank is set to 2, and may satisfy a condition of

$X,{Y\; \varepsilon {\left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2}}} \right\}.}}$

The Rank 2 precoding matrix may further include a precoding matrixgenerated when positions of individual rows of the first type precodingmatrix are changed.

The Rank 2 precoding matrix may further include a second type precodingmatrix configured in a form of

$\begin{bmatrix}1 & 0 \\0 & 1 \\X & 0 \\0 & Y\end{bmatrix},$

and a third type precoding matrix configured in a form of

$\begin{bmatrix}1 & 0 \\0 & 1 \\0 & Y \\X & 0\end{bmatrix},$

where individual rows of the precoding matrix may respectivelycorrespond to four antennas of the multiple antennas, and individualcolumns may respectively correspond to layers.

The Rank 2 precoding matrix may further include a precoding matrixgenerated when positions of individual columns of the first typeprecoding matrix are changed.

The codebook may include a first type precoding matrix, wherein thefirst type precoding matrix, serving as a Rank 3 precoding matrixutilized when the number of the multiple antennas is 4 and a rank is setto 3, is configured in a form of

${\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\X & 0 & 0\end{bmatrix}.},$

and satisfies a condition of

$X \in {\left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2}}} \right\}.}$

The Rank 3 precoding matrix may further include a precoding matrixgenerated when positions of individual rows of the first type precodingmatrix are changed. The Rank 3 precoding matrix may further include aprecoding matrix generated when positions of individual columns of thefirst type precoding matrix are changed. That is, the codebook mayincludes a precoding matrix configured to alternatively map a firstlayer to first and second antennas and second and third layers to thirdand fourth antennas, respectively, as the precoding matrix used for thecase when the number of antennas is 4 and the rank is 3.

When the number of antennas is 4, the Rank is 3, and the number ofcodewords is 2, one of the codeword is mapped to a single layer, and theother codeword is mapped to two layers. The precoding matrix can beconfigured so that the total transmission power from the layerperspective may be different in order to enforce uniform transmissionpower between multiple antennas. In such a case the precoding matrixcolumn which has larger effective transmission power is mapped to thelayer which is solely mapped to a single codeword. Thus in case ofprecoding matrix in the form of

$\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\X & 0 & 0\end{bmatrix},$

the first column is mapped to the layer which is solely mapped to asingle codeword, and the second and third column is mapped to layerswhich is mapped to the other codeword.

The codebook may include a different number of precoding matrices foreach rank.

Each of the uplink signals may be entered in units of a codeword, andthe mapping step of the uplink signals to the predetermined number oflayers may periodically change a layer mapped to a specific codeword toanother layer. One example of this periodicity can be 1 SC-FDMA symbol.

In another aspect of the present invention, a user equipment (UE) fortransmitting uplink signals via multiple antennas includes multipleantennas for transmitting and receiving signals, a memory for storing acodebook having a precoding matrix established in a manner that onelayer signal is transmitted to the multiple antennas, and a processorconnected to the multiple antennas and the memory so as to process theuplink signal transmission. The processor includes a layer mapper formapping the uplink signals to a predetermined number of layerscorresponding to a specific rank, a Discrete Fourier Transform (DFT)module for performing DFT spreading upon each of the predeterminednumber of layer signals, a precoder for precoding each of the DFT-spreadlayer signals received from the DFT module by selecting a specificprecoding matrix established in a manner that one layer signal istransmitted to each of the multiple antennas from among a codebookstored in the memory, and a transmission module for performing apredetermined process for constructing a Single Carrier-FrequencyDivision Multiple Access (SC-FDMA) symbol upon the precoded signals, andtransmitting the processed signals to a base station (BS) via themultiple antennas.

In this case, the memory may store the codebook. The processor mayperform the antenna shift and/or the layer shift either in a differentway from the precoding of a precoder or through row permutation and/orcolumn permutation of a precoding matrix.

It is to be understood that both the foregoing general description andthe following detailed description of the present invention areexemplary and explanatory and are intended to provide furtherexplanation of the invention as claimed.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are included to provide a furtherunderstanding of the invention and are incorporated in and constitute apart of this application, illustrate embodiment(s) of the invention andtogether with the description serve to explain the principle of theinvention. In the drawings:

FIG. 1 is a conceptual diagram illustrating a general MIMO communicationsystem.

FIGS. 2 and 3 illustrate a general structure of a transmitter based on aMIMO technology.

FIG. 4 is a conceptual diagram illustrating a method for precodinginformation of each layer and transmitting the precoded information viaan antenna.

FIG. 5 is a conceptual diagram illustrating a general SC-FDMA scheme.

FIG. 6 is a conceptual diagram illustrating a method for mapping acodeword to several layers.

FIG. 7 is a conceptual diagram illustrating a method for performing aDFT upon each layer after performing codeword-to-layer mapping (i.e.,codeword-layer mapping) so as to prevent a CM value for each antennafrom being increased.

FIG. 8 is a conceptual diagram illustrating a method for performingpermutation on the position of a row or column of a precoding matrix.

FIG. 9 is a conceptual diagram illustrating a chordal distance.

FIG. 10 is a block diagram illustrating a general base station (BS) anda general user equipment (UE).

FIGS. 11 and 12 illustrate an SC-FDMA scheme for transmitting an uplinksignal in a 3GPP LTE system and an OFDMA scheme for transmitting adownlink signal in the 3GPP LTE system.

FIG. 13 is a block diagram illustrating a processor for enabling a basestation (BS) to transmit a downlink signal using a MIMO scheme in a 3GPPLTE system.

FIG. 14 illustrates a processor of a UE according to one embodiment ofthe present invention.

DETAILED DESCRIPTION OF THE INVENTION

Reference will now be made in detail to the preferred embodiments of thepresent invention, examples of which are illustrated in the accompanyingdrawings. Wherever possible, the same reference numbers will be usedthroughout the drawings to refer to the same or like parts.

The detailed description, which will be given below with reference tothe accompanying drawings, is intended to explain exemplary embodimentsof the present invention, rather than to show the only embodiments thatcan be implemented according to the present invention. The followingdetailed description includes specific details in order to provide athorough understanding of the present invention. However, it will beapparent to those skilled in the art that the present invention may bepracticed without such specific details. For example, the followingdescription will be given centering on specific terms, but the presentinvention is not limited thereto and any other terms may be used torepresent the same meanings. Also, wherever possible, the same referencenumbers will be used throughout the drawings to refer to the same orlike parts.

Peak power to Average Power Ratio (PAPR) is a parameter indicatingcharacteristics of a waveform. PAPR is a specific value acquired when apeak amplitude of the waveform is divided by a time-averaged Root MeanSquare (RMS) value of the waveform. PAPR is a dimensionless value. Ingeneral, a PAPR of a single carrier signal is better than that of amulti-carrier signal.

An LTE-Advanced scheme can implement MIMO technology using SingleCarrier-Frequency Division Multiple Access (SC-FDMA) so as to maintain asuperior CM property. When using general precoding, a signal includinginformation corresponding to several layers is multiplexed andtransmitted via a single antenna, so that the signal transmitted viathis antenna may be considered to be a kind of multi-carrier signal.PAPR is associated with a dynamic range that must be supported by apower amplifier of a transmitter, and a CM value is another valuecapable of being used as a substitute for the PAPR.

FIG. 2 shows a general structure of a transmitter based on a MIMOtechnology.

In FIG. 2, one or more codewords are mapped to a plurality of layers. Inthis case, mapping information is mapped to each physical antenna by aprecoding process, and is then transmitted via each physical antenna.

FIG. 3 is a detailed block diagram illustrating the MIMO-basedtransmitter shown in FIG. 2.

The term ‘codeword’ indicates that Cyclic Redundancy Check (CRC) bitsare attached to data information and are then encoded by a specificcoding method. There are a variety of coding methods, for example, aturbo code, a tail biting convolution code, and the like. Each codewordis mapped to one or more layers (i.e., one or more virtual layers), anda total number of mapped layers is equal to a rank value. In otherwords, if a transmission rank is 3, a total number of transmissionlayers is also set to 3. Information mapped to each layer is precoded.In this case, data information mapped to each layer is mapped to aphysical layer through a precoding process (where, the term ‘layer’means a virtual layer as far as it especially designates a physicallayer). Information is transmitted to each antenna via each physicallayer. Under the condition that no specified explanation is shown inFIG. 3, the precoding is carried out in a frequency domain, and an OFDMinformation transmission scheme is used for information mapped to thephysical layer. The information mapped to the physical layer is mappedto a specific frequency domain, and is then IFFT-processed. After that,a cyclic prefix (CP) is attached to the IFFT result. Thereafter,information is transmitted to each antenna via a radio frequency (RF)chain.

The precoding process may be carried out by matrix multiplication. Ineach of the matrices, the number of rows is equal to the number ofphysical layers (i.e., the number of antennas), and the number ofcolumns is equal to a rank value. The rank value is equal to the numberof layers, so that the number of columns is equal to the number oflayers. Referring to the following equation 2, information mapped to alayer (i.e., a virtual layer) is x₁ and x₂, each element p_(ij) of a(4×2) matrix is a weight used for precoding. y₁, y₂, y₃, and y₄ areinformation mapped to physical layers, and are transmitted viarespective antennas using individual OFDM transmission schemes.

$\begin{matrix}{\begin{bmatrix}y_{1} \\y_{2} \\y_{3} \\y_{4}\end{bmatrix} = {\begin{bmatrix}p_{11} & p_{21} \\p_{12} & p_{22} \\p_{13} & p_{23} \\p_{14} & p_{24}\end{bmatrix} \cdot \begin{bmatrix}x_{1} \\x_{2}\end{bmatrix}}} & \left\lbrack {{Equation}\mspace{14mu} 2} \right\rbrack\end{matrix}$

In the following description, a virtual layer will hereinafter bereferred to as a layer so long as such use will not lead to confusion.An operation for mapping a virtual layer signal to a physical layer willhereinafter be considered to be an operation for directly mapping alayer to an antenna.

The precoding method can be mainly classified into two methods, i.e., awideband precoding method and a subband precoding method.

The wideband precoding method is as follows. According to the widebandprecoding method, when precoding is carried out in a frequency domain,the same precoding matrix is applied to all information transmitted tothe frequency domain.

FIG. 4 is a conceptual diagram illustrating a method for precodinginformation of each layer and transmitting the precoded information viaan antenna.

Referring to FIG. 4, it can be recognized that information correspondingto a plurality of layers is precoded while being classified according tosubcarriers of each frequency domain, and the precoded information istransmitted via each antenna. All precoding matrices ‘P’ for use in thewideband precoding method are equal to each other.

The subband precoding method is provided by the extension of thewideband precoding method. The subband precoding method applies avariety of precoding matrices to each subcarrier without applying thesame precoding matrix to all subcarriers. In other words, according tothe subband precoding method, a precoding matrix ‘P’ is used in aspecific subcarrier, and another precoding matrix ‘M’ is used in theremaining subcarriers other than the specific subcarrier. Herein,element values of the precoding matrix ‘P’ are different from those ofthe other precoding matrix ‘M’.

Uplink signal transmission is relatively sensitive to PAPR or CMproperties as compared to downlink signal transmission. The increase offilter costs caused by the increase of PAPR or CM properties maygenerate more serious problems in a user equipment (UE). Thus, theSC-FDMA scheme is used for uplink signal transmission.

FIG. 5 is a conceptual diagram illustrating a general SC-FDMA scheme.

As shown in FIG. 5, the OFDM scheme and the SC-FDMA scheme areconsidered to be identical with each other, because they convert aserial signal into parallel signals, map the parallel signals tosubcarriers, perform an IDFT or IFFT process on the mapped signals,convert the IDFT- or IFFT-processed signals into a serial signal, attacha cyclic prefix (CP) to the resultant serial signal, and transmit the CPresultant signal via a radio frequency (RF) module. However, in contrastto the OFDM scheme, the SC-FDMA scheme converts parallel signals into aserial signal, and performs DFT spreading upon the serial signal, sothat it reduces the influence of a next IDFT or IFFT process andmaintains a single signal characteristic of more than a predeterminedlevel.

In the meantime, the reason why the CM value is degraded when a MIMOscheme is applied to uplink signal transmission is as follows. If aplurality of single-carrier signals each having good CM properties issimultaneously overlapped with each other, the overlapped signals mayhave poor CM properties. Therefore, if the SC-FDMA system multiplexesoutput information of several layers using a minimum number ofsingle-carrier signals or one single-carrier signal on a single physicalantenna, a transmission signal having a good CM can be generated.

A codeword-layer mapping process may be performed before information tobe transmitted is precoded. Since the SC-FDMA scheme is generally usedfor one transmission mode (1Tx), the number of layers is 1. However, ifthe SC-FDMA scheme supports a MIMO scheme, the number of layers isplural, and a codeword composed of a single transport block may bemapped to a plurality of layers.

FIG. 6 is a conceptual diagram illustrating a method for mapping acodeword to several layers.

Referring to FIG. 6, if the codeword-layer mapping is carried out aftera DFT process for the SC-FDMA scheme is performed, a CM value may beincreased. That is, because an output signal of a DFT block undergoesother processes before entering an IFFT module, i.e., because the outputsignal of the DFT block is divided into two layers, a CM value may beincreased.

FIG. 7 is a conceptual diagram illustrating a method for performing DFTupon each layer after performing codeword-to-layer mapping (i.e., acodeword-layer mapping) so as to prevent a CM value for each antennafrom being increased.

Therefore, if the number of DFT blocks is changed while being classifiedaccording to layer numbers based on a rank value, a low CM value can bemaintained. That is, the output signal of the DFT block is directlyinput to the IFFT block without passing through other processes, so thata low CM value can be maintained. In the case of actual implementation,a plurality of layers may share a single DFT block.

If a plurality of layer signals is transmitted via a single antenna byapplying the MIMO scheme to uplink signal transmission, a PAPR or a CMproperty may be deteriorated. In order to overcome the above-mentionedproblem, the following embodiments of the present invention willdescribe a method for designing a codebook based on a precoding matrixby which only one layer signal is transmitted via a single antenna.

For convenience of description and better understanding of the presentinvention, in a transmission system, it is assumed that a set of signalstransferred to a precoding block is set to ‘x’, and a set of precodedsignals is set to ‘y’. In this case, if the precoding matrix is ‘P’, thefollowing equation 3 is acquired.

Y=P·x  [Equation 3]

In Equation 3, a dimension of ‘P’ is N_(T)×N_(L), a dimension of ‘x’ isN_(L)×1, a dimension of ‘y’ is N_(T)×1. In this case, N_(T) is thenumber of antennas, and N_(L) is the number of layers.

In the following description, a principle of designing a codebook thatis capable of being applied to uplink signal transmission using a MIMOscheme by a UE will be firstly described in chapter (I), and a detailedformat of the codebook will then be described in chapter (II).

I. Principle of Codebook Design

<2Tx Codebook>

A variety of embodiments according to a structure of a precoding matrixcontained in a codebook used in a 2Tx mode will hereinafter bedescribed.

The method according to embodiments of the present invention includes:generating a plurality of streams by mapping a codeword to a pluralityof layers; and precoding the generated streams, mapping the precodedstreams to a plurality of antennas, and transmitting the mapped resultvia the antennas. In this case, the codebook may be configured asfollows. A precoding matrix used in Rank 1 and the other precodingmatrix used in Rank 2 will be described in different ways.

2Tx-Rank 1 Precoding Matrix

In case of 2Tx-Rank 1, Equation 3 can be rewritten as the followingequation 4 according to embodiments of the present invention.

$\begin{matrix}{y = {\begin{bmatrix}y_{1} \\y_{2}\end{bmatrix} = {{P \cdot x} = {{\begin{bmatrix}a \\b\end{bmatrix} \cdot \left\lbrack x_{1} \right\rbrack} = \begin{bmatrix}{ax}_{1} \\{bx}_{1}\end{bmatrix}}}}} & \left\lbrack {{Equation}\mspace{14mu} 4} \right\rbrack\end{matrix}$

In general, if it is assumed that a wideband precoding scheme is used, aspecific constant value is multiplied by a signal of each layeraccording to a Rank 1 precoding scheme, a PAPR and CM value of a signaltransmitted via each antenna in the 2Tx mode are equal to those in a 1Txmode. Thus, when using wideband precoding, the PAPR and the CM are notaffected by values of a 2Tx-Rank 1 precoding matrix.

Precoding is a method for changing a channel so as to acquire aconstructive effect between signals transferred via each channel. Thus,transmission performance of each signal is improved. Accordingly, ‘a’indicating a first element of the precoding matrix P illustrated inEquation 4 is set to ‘1’, and a second element ‘b’ of the precodingmatrix P may be set to an arbitrary value. Signals transferred viarespective antennas have the same power, so that all power amplifierscontained in each antenna can be maximally used. For this purpose, theabove-mentioned second element ‘b’ may be a complex number having anabsolute value of 1. In other words, P shown in Equation 4 may berepresented by

$P = \begin{bmatrix}1 \\^{j\theta}\end{bmatrix}$

There is a limitation in the number of precoding matrices contained in acodebook used for the precoding, because both a transmission end and areception end must have a codebook and information about a predeterminedprecoding matrix is communicated between the transmission end and thereception end. As a result, the transmission end and the reception endmust use a limited number of precoding matrices. For this operation, acomplex number that has an absolute value of 1 and a phase correspondingto any one of +0°, +45°, +90°, +135°, +180°, −135°, −90°, and −45° maybe used as each element of the precoding matrix. That is, in theabove-mentioned expression,

$P = \begin{bmatrix}1 \\^{j\theta}\end{bmatrix}$

θ may be represented by

$\theta \in {\left\{ {0,\frac{\pi}{4},\frac{\pi}{2},\frac{3\pi}{4},\pi,\frac{5\pi}{4},\frac{6\pi}{4},\frac{7\pi}{4}} \right\}.}$

In other words, P may be represented by

$P \in {\left\{ {\begin{bmatrix}1 \\1\end{bmatrix},\begin{bmatrix}1 \\\frac{1 + j}{\sqrt{2}}\end{bmatrix},\begin{bmatrix}1 \\j\end{bmatrix},\begin{bmatrix}1 \\\frac{1 - j}{\sqrt{2}}\end{bmatrix},\begin{bmatrix}1 \\{- 1}\end{bmatrix},\begin{bmatrix}1 \\\frac{{- 1} - j}{\sqrt{2}}\end{bmatrix},\begin{bmatrix}1 \\{- j}\end{bmatrix},\begin{bmatrix}1 \\\frac{{- 1} + j}{\sqrt{\square}}\end{bmatrix}} \right\}.}$

2Tx-Rank 2 Precoding Matrix

In case of a 2Tx-Rank 2, the equation 3 may be rewritten as thefollowing equation 5.

$\begin{matrix}{y = {\begin{bmatrix}y_{1} \\y_{2}\end{bmatrix} = {{P \cdot x} = {{\begin{bmatrix}p_{11} & p_{12} \\p_{21} & p_{22}\end{bmatrix} \cdot \begin{bmatrix}x_{1} \\x_{2}\end{bmatrix}} = \begin{bmatrix}{{p_{11}x_{1}} + {p_{12}x_{2}}} \\{{p_{21}x_{1}} + {p_{22}x_{2}}}\end{bmatrix}}}}} & \left\lbrack {{Equation}\mspace{14mu} 5} \right\rbrack\end{matrix}$

In Equation 5, the signal y_(k) transferred via each antenna is composedof a combination of several input signals x_(i), so that a CM value maybe increased.

In this case, if each of p₁₂ and p₂₁ is set to zero ‘0’ or if each ofp₁₁ and p₂₂ is set to zero ‘0’, only one signal can be transmitted viaeach antenna. Thus, if it is assumed that a CM value of a signal x_(i)is considered to be good, a CM value of the precoded signal also becomesgood. In association with FIG. 7, in the case where a codeword is mappedto each layer, DFT spreading is applied to the resultant signal mappedto each layer, and a precoding process for allowing each antenna totransmit only one layer signal is carried out, the same effect as in anIDFT or IFFT process that is performed as soon as a DFT process wasperformed can be acquired, and a PAPR or CM property can be maintainedat a good status. A detailed description of this will hereinafter beexplained in the following description.

In this case, if each of p₁₂ and p₂₁ is zero ‘0’, a signal correspondingto each layer is transmitted via each antenna after being multiplied bya constant complex value. As a result, although the above-mentionedconstant complex value is set to 1, performance is not affected by thisconstant complex value of 1.

Therefore, Equation 5 can be represented by the following equation 6.

$\begin{matrix}{{y = {\begin{bmatrix}y_{1} \\y_{2}\end{bmatrix} = {{P \cdot x} = {{\begin{bmatrix}p_{11} & 0 \\0 & p_{22}\end{bmatrix} \cdot \begin{bmatrix}x_{1} \\x_{2}\end{bmatrix}} = {\begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix} \cdot \begin{bmatrix}x_{1} \\x_{2}\end{bmatrix}}}}}},\mspace{79mu} {P \in \left\{ \begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix} \right\}}} & \left\lbrack {{Equation}\mspace{14mu} 6} \right\rbrack\end{matrix}$

<4Tx Codebook>

A variety of embodiments according to a structure of a precoding matrixcontained in a codebook used in a 4Tx mode will hereinafter bedescribed.

The method according to embodiments of the present invention includes:generating a plurality of streams by mapping a codeword to a pluralityof layers; and precoding the generated streams, mapping the precodedstreams to a plurality of antennas, and transmitting the mapped resultvia the antennas. In this case, the codebook may be configured asfollows. Precoding matrices respectively used in Rank 1, Rank 2, Rank 3,and Rank 4 will be described in different ways.

4Tx-Rank 1 Precoding Matrix

In case of 4Tx-Rank 1, Equation 3 can be rewritten as the followingequation 7.

$\begin{matrix}{y = {\begin{bmatrix}y_{1} \\y_{2} \\y_{3} \\y_{4}\end{bmatrix} = {{P \cdot x} = {{\begin{bmatrix}a \\b \\c \\d\end{bmatrix} \cdot \left\lbrack x_{1} \right\rbrack} = \begin{bmatrix}{ax}_{1} \\{bx}_{1} \\{cx}_{1} \\{dx}_{1}\end{bmatrix}}}}} & \left\lbrack {{Equation}\mspace{14mu} 7} \right\rbrack\end{matrix}$

In case of using the wideband precoding scheme in the same manner as inthe 2Tx-Rank 1 codebook, a CM of a signal transmitted via each antennaby a 4Tx-Rank 1 precoding process is equal to a CM of a signal used inthe 1Tx mode. Thus, all kinds of precoding matrices can be freelyapplied to such a CM without any problems.

4Tx-Rank 2 Precoding Matrix

In case of 4Tx-Rank 2, Equation 3 can be rewritten as the followingequation 8.

$\begin{matrix}{y = {\begin{bmatrix}y_{1} \\y_{2} \\y_{3} \\y_{4}\end{bmatrix} = {{P \cdot x} = {{\begin{bmatrix}p_{11} & p_{12} \\p_{21} & p_{22} \\p_{31} & p_{32} \\p_{41} & p_{42}\end{bmatrix} \cdot \begin{bmatrix}x_{1} \\x_{2}\end{bmatrix}} = \begin{bmatrix}{{p_{11}x_{1}} + {p_{12}x_{2}}} \\{{p_{21}x_{1}} + {p_{22}x_{2}}} \\{{p_{31}x_{1}} + {p_{32}x_{2}}} \\{{p_{41}x_{1}} + {p_{42}x_{2}}}\end{bmatrix}}}}} & \left\lbrack {{Equation}\mspace{14mu} 8} \right\rbrack\end{matrix}$

In a 4Tx-Rank 2 codebook, in a similar way as in the 2Tx-Rank 2codebook, a specific element of a precoding matrix is set to zero ‘0’,so that the overlapping of signals transmitted via respective antennasis minimized and thus a CM can be maintained at a low value.

In Equation 8, if it is assumed that p_(k1) or p_(k2) in a signal(p_(k1)x₁+p_(k2)x₂) transmitted via each antenna is set to zero ‘0’, thesignal transmitted via each antenna becomes equal to a signaltransmitted from a single layer, and therefore a CM of the signaltransmitted via each antenna can be maintained at a low value.

In one embodiment of the present invention, ‘P’ included in Equation 8may be represented by

$P = {\begin{bmatrix}p_{11} & 0 \\p_{21} & 0 \\0 & p_{32} \\0 & p_{42}\end{bmatrix}.}$

Equation 8 may be rewritten as the following equation 9.

$\begin{matrix}{y = {\begin{bmatrix}y_{1} \\y_{2} \\y_{3} \\y_{4}\end{bmatrix} = {{P \cdot x} = {{\begin{bmatrix}p_{11} & 0 \\p_{21} & 0 \\0 & p_{32} \\0 & p_{42}\end{bmatrix} \cdot \begin{bmatrix}x_{1} \\x_{2}\end{bmatrix}} = \begin{bmatrix}{p_{11}x_{1}} \\{p_{21}x_{2}} \\{p_{32}x_{2}} \\{p_{42}x_{2}}\end{bmatrix}}}}} & \left\lbrack {{Equation}\mspace{14mu} 9} \right\rbrack\end{matrix}$

Referring to Equation 9, only one layer is mapped to a signaltransmitted via each antenna. From the viewpoint of a single layer, itis considered that the 2Tx-Rank 1 precoding is applied to informationtransmitted via this single layer. Thus, the 4Tx-Rank 2 precoding matrixcan be configured using a 2Tx-Rank 2 precoding matrix. In other words,the 4Tx-Rank 2 precoding matrix may be a super matrix of the 2Tx-Rank 1precoding matrix.

For example, ‘P’ according to one embodiment of the present inventioncan be represented by Equation 10.

$\begin{matrix}{\mspace{79mu} {{P = {\begin{bmatrix}\begin{bmatrix}1 \\X\end{bmatrix} & 0 \\0 & 0 \\0 & \begin{bmatrix}1 \\Y\end{bmatrix}\end{bmatrix} = \begin{bmatrix}1 & 0 \\X & 0 \\0 & 1 \\0 & Y\end{bmatrix}}},X,{Y \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2}}} \right\}}}} & \left\lbrack {{Equation}\mspace{14mu} 10} \right\rbrack\end{matrix}$

The above-mentioned 2Tx-Rank1 precoding matrix is used for a method fortransmitting information by applying two antennas to a single layersignal. However, if it is assumed that there are four physical antennas,communication performance may be changed according to which combinationcomposed of two antennas is used for data transmission. In this case,the selected combination of antennas may be changed according to a valueof the precoding matrix P.

For example, according to one embodiment of the present invention, theprecoding matrix P may be configured in various formats. Respectiveformats may indicate different antenna combinations.

$\begin{matrix}{P \in \left\{ {\begin{bmatrix}1 & 0 \\X & 0 \\0 & 1 \\0 & Y\end{bmatrix},\begin{bmatrix}1 & 0 \\0 & 1 \\X & 0 \\0 & Y\end{bmatrix},\begin{bmatrix}1 & 0 \\0 & 1 \\0 & Y \\X & 0\end{bmatrix}} \right\}} & \left\lbrack {{Equation}\mspace{14mu} 11} \right\rbrack\end{matrix}$

In Equation 11, if an appropriate value is selected as a precodingmatrix P, performance improvement due to precoding can be enhanced. Ifthe precoding matrix is configured as described above, a signalcorresponding to each layer uses two antennas among a total of fourantennas, channel estimation performances among respective layers becomesimilar to each other, and a CM value for each antenna can be minimized.

Generally, although a constant value is multiplied by a specific columnvector of an arbitrary precoding matrix, characteristics of theprecoding matrix are not changed. Therefore, although a constant valueis multiplied by a specific column of the above-mentioned precodingmatrix, characteristics of the precoding matrix are not changed. As aresult, the above-mentioned operation for multiplying a constant valueby a specific column vector of the precoding matrix does not depart fromthe scope of the present invention.

In addition, if a predetermined scaling factor is multiplied by theprecoding matrix shown in Equation 11, the multiplied result may berepresented by the following equation 12.

$\begin{matrix}{\mspace{79mu} {{P \in \left\{ {{k \cdot \begin{bmatrix}1 & 0 \\X & 0 \\0 & 1 \\0 & Y\end{bmatrix}},{k \cdot \begin{bmatrix}1 & 0 \\0 & 1 \\X & 0 \\0 & Y\end{bmatrix}},{k \cdot \begin{bmatrix}1 & 0 \\0 & 1 \\0 & Y \\X & 0\end{bmatrix}}} \right\}}{X,{Y \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2}}} \right\}}}}} & \left\lbrack {{Equation}\mspace{14mu} 12} \right\rbrack\end{matrix}$

4Tx-Rank 3 Precoding Matrix (1)

In case of 4Tx-Rank 3, Equation 3 can be rewritten as the followingequation 13.

$\begin{matrix}{y = {\begin{bmatrix}y_{1} \\y_{2} \\y_{3} \\y_{4}\end{bmatrix} = {{P \cdot x} = {{\begin{bmatrix}p_{11} & p_{12} & p_{13} \\p_{21} & p_{22} & p_{23} \\p_{31} & p_{32} & p_{33} \\p_{41} & p_{42} & p_{43}\end{bmatrix} \cdot \begin{bmatrix}x_{1} \\x_{2} \\x_{3}\end{bmatrix}} = {\quad\begin{bmatrix}{{p_{11}x_{1}} + {p_{12}x_{2}} + {p_{13}x_{3}}} \\{{p_{21}x_{1}} + {p_{22}x_{2}} + {p_{23}x_{3}}} \\{{p_{31}x_{1}} + {p_{32}x_{2}} + {p_{33}x_{3}}} \\{{p_{41}x_{1}} + {p_{42}x_{2}} + {p_{43}x_{3}}}\end{bmatrix}}}}}} & \left\lbrack {{Equation}\mspace{14mu} 13} \right\rbrack\end{matrix}$

In a 4Tx-Rank 3 precoding matrix in a similar way as in the 4Tx-Rank 2precoding matrix, a specific element of a precoding matrix is set tozero ‘0’, so that the overlapping of signals transmitted via respectiveantennas is minimized and thus a CM can be maintained at a low value.

In Equation 13, if it is assumed that p_(k1), p_(k2), or p_(k3) in asignal (p_(k1)x₁+p_(k2)x₂+p_(k3)x₃) transmitted via each antenna is setto zero ‘0’, a CM of the signal transmitted via each antenna can bemaintained at a low value.

In one embodiment of the present invention, ‘P’ included in Equation 12may be represented by

$P = {\begin{bmatrix}p_{11} & 0 & 0 \\0 & p_{22} & 0 \\0 & 0 & p_{33} \\p_{41} & p_{42} & P_{43}\end{bmatrix}.}$

Equation 13 may be rewritten as the following equation 14.

$\begin{matrix}{y = {\begin{bmatrix}y_{1} \\y_{2} \\y_{3} \\y_{4}\end{bmatrix} = {{P \cdot x} = {{\begin{bmatrix}p_{11} & 0 & 0 \\0 & p_{22} & 0 \\0 & 0 & p_{33} \\p_{41} & p_{42} & P_{43}\end{bmatrix} \cdot \begin{bmatrix}x_{1} \\x_{2} \\x_{3}\end{bmatrix}} = {\quad\begin{bmatrix}{p_{11}x_{1}} \\{p_{22}x_{2}} \\{p_{33}x_{3}} \\{{p_{41}x_{1}} + {p_{42}x_{2}} + {p_{43}x_{3}}}\end{bmatrix}}}}}} & \left\lbrack {{Equation}\mspace{14mu} 14} \right\rbrack\end{matrix}$

In Rank 3, the number of layers to be transmitted is 3, and the numberof physical antennas is 4. In this case, each of the three antennas maybe independently mapped to a single layer. Herein, only a signal of asingle layer may be mapped to the remaining one antenna, or signals ofat least two layers may be mapped to the remaining one antenna. If onlya signal of a specific single layer is mapped to the remaining oneantenna, a CM of the signal transmitted via this antenna may have goodcharacteristics, but communication performance of the specific singlelayer may be different from that of another layer. For example, in thecase where information of a first layer (Layer 1) is mapped to a firstantenna (Antenna 1) and a fourth antenna (Antenna 4), information of asecond layer (Layer 2) is mapped to a second antenna (Antenna 2), andinformation of a third layer (Layer 3) is mapped to a third antenna(Antenna 3), communication performance of the Layer 1 information may bedifferent from that of either the Layer 2 or the Layer 3.

In one embodiment of the present invention, in order to minimize a CMvalue for each antenna in the precoding process, the precoding matrix Pmay have any one of the values of P₁, P₂, and P₃ shown in the followingequation 15.

$\begin{matrix}{{{P_{1} = \begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\X & 0 & 0\end{bmatrix}},{P_{2} = \begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\0 & Y & 0\end{bmatrix}},{P_{3} = \begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\0 & 0 & Z\end{bmatrix}}}\mspace{79mu} {{where},X,Y,{Z \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2}}} \right\}}}} & \left\lbrack {{Equation}\mspace{14mu} 15} \right\rbrack\end{matrix}$

In case of using the above-mentioned precoding matrices P₁, P₂, and P₃,numbers of antennas used for individual layers are different from eachother. However, if it is assumed that the precoding matrices P₁, P₂ andP₃ are evenly used to transmit certain information, instead of using anyone of the precoding matrices P₁, P₂ and P₃, numbers of antennas usedfor individual layers may be normalized. Although the precoding matricesP₁, P₂ and P₃ can be alternately used in a frequency domain, a singlecarrier property of a signal composed of a single carrier is damaged, sothat a CM value is unavoidably increased. Therefore, if the precodingmatrices P₁, P₂ and P₃ are alternately applied to each SC-FDMA symbol,no additional increase in CM is achieved. In case of transmitting data,information may be decoded in units of one subframe. Thus, if theprecoding matrices P₁, P₂ and P₃ are alternately applied to each SC-FDMAsymbol, each layer information of the whole information transmitted viaa single subframe can be transmitted via the same number of antennas onaverage.

In another embodiment of the present invention, the position of anantenna used by each layer is changed so that performance can beimproved. The changing of the antenna position may be carried out withtime. In particular, the antenna position can be changed at each SC-FDMAsymbol. A detailed method for changing the antenna position willhereinafter be described in detail.

For example, the position of a value other than ‘O’ in the precodingmatrix is changed to another position in the range of a row vector, sothat the position of an antenna via which each layer signal istransmitted can be changed to another position. As another example, theabove-mentioned method may be implemented by a row/column permutationbecause position permutation is carried out between rows or columns of agiven precoding matrix.

FIG. 8 is a conceptual diagram illustrating a method for performingpermutation on the position of a row or column of a precoding matrix.

In more detail, FIG. 8(a) is a conceptual diagram illustrating a methodfor performing permutation on the position of a row, and FIG. 8(b) is aconceptual diagram illustrating a method for performing permutation onthe position of a column.

In the precoding matrix shown in Equation 15, a precoding matrix P₁ canbe row-permuted and/or column-permuted, so that a precoding matrix P₂ orP₃ can be generated. Therefore, in the structure such as the precodingmatrix P₁, P₂ or P₃, a new unique precoding matrix can be acquired onlyby row permutation.

The order of rows changed by row permutation available in the 4Tx modecan be represented by the following expression.

{1, 2, 3, 4}, {1, 2, 4, 3}, {1, 3, 2, 4}, {1, 3, 4, 2},

{1, 4, 2, 3}, {1, 4, 3, 2}, {2, 1, 3, 4}, {2, 1, 4, 3},

{2, 3, 1, 4}, {2, 3, 4, 1}, {2, 4, 1, 3}, {2, 4, 3, 1},

{3, 2, 1, 4}, {3, 2, 4, 1}, {3, 1, 2, 4}, {3, 1, 4, 2},

{3, 4, 2, 1}, {3, 4, 1, 2}, {4, 2, 3, 1}, {4, 2, 1, 3},

{4, 3, 2, 1}, {4, 3, 1, 2}, {4, 1, 2, 3}, {4, 1, 3, 2}

In the above-mentioned expression, {w, x, y, z} means that row vectors1, 2, 3 and 4 of the precoding matrix are rearranged in the order ofparenthesized numbers on the condition that a given precoding matrixP_(k) exists.

By row permutation, signals corresponding to a specific layer are mappedto different antennas. By column permutation, the same effect as in theswitching of information of different layers can be acquired. If thereis no need to distinguish performance of each layer, and a system forrequesting similar performance from each layer need not utilize thecolumn permutation. Thus, the effect corresponding to antenna selectioncan be acquired using only the row permutation.

In the meantime, in the case where a given scaling factor is multipliedby each precoding matrix shown in Equation 15, the result can berepresented by the following equation 16.

$\begin{matrix}{{P_{1} = {{{k \cdot \begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\X & 0 & 0\end{bmatrix}}\; P_{2}} = {{{k \cdot \begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\0 & Y & 0\end{bmatrix}}\; P_{3}} = {k \cdot \begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\0 & 0 & Z\end{bmatrix}}}}},X,Y,{Z \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2}}} \right\}}} & \left\lbrack {{Equation}\mspace{14mu} 16} \right\rbrack\end{matrix}$

4Tx-Rank 3 Precoding Matrix (2)

In case of 4Tx-Rank 3, if each antenna transmits informationcorresponding to only one layer, a CM value of a signal transmitted viaeach antenna can be maintained at a low value, however information ofonly one layer is transmitted via only one antenna so that communicationperformance can be deteriorated. Therefore, in case of 4Tx-Rank 3, thereis a need for a codebook to be designed in a manner that a maximum oftwo layers are multiplexed and transmitted via a single antenna, so thatthe increment of CM can be minimized and at the same time communicationperformance can be increased.

In accordance with one embodiment of the present invention, wheninformation corresponding to two layers is transmitted via a singleantenna, the precoding matrix P shown in Equation 13 can be representedby P₄ of Equation 17 or P₅ of Equation 18.

$\begin{matrix}{\mspace{79mu} {{P_{4} = \begin{bmatrix}1 & 0 & 1 \\X & 0 & Z \\0 & 1 & 0 \\0 & Y & 0\end{bmatrix}},{X \neq Z},X,Y,{Z \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2}}} \right\}}}} & \left\lbrack {{Equation}\mspace{14mu} 17} \right\rbrack \\{\mspace{79mu} {{P_{5} = \begin{bmatrix}1 & 0 & 0 \\X & 1 & 0 \\0 & Y & 1 \\0 & 0 & Z\end{bmatrix}},X,Y,{Z \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2}}} \right\}}}} & \left\lbrack {{Equation}\mspace{14mu} 18} \right\rbrack\end{matrix}$

In Equation 17, in order to satisfy Rank 3, ‘X’ must be different from‘Z’ in the precoding matrix P₄.

A method for using the precoding matrix P₄ or P₅ has a disadvantage inthat only a signal of a single layer is transmitted via the otherantenna whereas a signal of two layers is multiplexed and transmittedvia a specific antenna.

In one embodiment of the present invention, in order to obviate theabove-mentioned problem, the precoding matrix P may have any one ofvalues Pa, P₇ and Pa shown in the following equation 19.

$\begin{matrix}{{{P_{6} = \begin{bmatrix}1 & 0 & Z \\X & 1 & 0 \\0 & Y & 1 \\A & 0 & C\end{bmatrix}},{P_{7} = \begin{bmatrix}1 & 0 & Z \\X & 1 & 0 \\0 & Y & 1 \\0 & B & C\end{bmatrix}},{P_{8} = \begin{bmatrix}1 & 0 & Z \\X & 1 & 0 \\0 & Y & 1 \\A & B & 0\end{bmatrix}}}{{where},X,Y,Z,A,B,{C \in \begin{Bmatrix}{1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},} \\{\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2}}}\end{Bmatrix}}}} & \left\lbrack {{Equation}\mspace{14mu} 19} \right\rbrack\end{matrix}$

In association with the precoding matrix P₄, P₅, P₆, P₇ or P₈, rowpermutation and/or column permutation can be carried out on the 4Tx-Rank3 precoding matrix. Because the row permutation and the columnpermutation are carried out, an antenna selection function and layerpermutation function for enabling a signal of a specific layer to betransmitted via an arbitrary antenna can be implemented by theprecoding.

In one embodiment of the present invention, individual column vectors ofthe precoding matrix may be configured to have orthogonalitytherebetween.

If individual column vectors of the precoding matrix are configured tohave orthogonality therebetween, the precoding matrix is able to satisfyproperties of a one side unitary matrix. That is, the precoding matrix Pcan have a characteristic denoted by the following equation 20.

P ^(H) P=α·I≠PP ^(H)  [Equation 20]

In one embodiment of the present invention, the precoding matrix of Rank3 can be configured in the form of the following equation 21. Theprecoding matrix P for satisfying the following equation 21 is able tosatisfy the relationship illustrated in Equation 20.

$\begin{matrix}{\mspace{79mu} {{P = \begin{bmatrix}1 & 0 & 1 \\X & 0 & {- X} \\0 & 1 & 0 \\0 & Y & 0\end{bmatrix}},X,{Y \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2}}} \right\}}}} & \left\lbrack {{Equation}\mspace{14mu} 21} \right\rbrack\end{matrix}$

In Equation 21, since the relationship denoted by

${P^{H}P} = {{\begin{bmatrix}1 & X^{*} & 0 & 0 \\0 & 0 & 1 & Y^{*} \\1 & {- X^{*}} & 0 & 0\end{bmatrix} \cdot \begin{bmatrix}1 & 0 & 1 \\X & 0 & {- X} \\0 & 1 & 0 \\0 & Y & 0\end{bmatrix}} = {\begin{bmatrix}2 & 0 & 0 \\0 & 2 & 0 \\0 & 0 & 2\end{bmatrix} = {\alpha \mspace{14mu} I}}}$

is satisfied, it can be recognized that the matrix P satisfies Equation20.

4Tx-Rank 4 Precoding Matrix (1)

In case of 4Tx-Rank 4, Equation 3 can be rewritten as the followingequation 22.

$\begin{matrix}{y = {\begin{bmatrix}y_{1} \\y_{2} \\y_{3} \\y_{4}\end{bmatrix} = {{P \cdot x} = {{\begin{bmatrix}p_{11} & p_{12} & p_{13} & p_{14} \\p_{21} & p_{22} & p_{23} & p_{24} \\p_{31} & p_{32} & p_{33} & p_{34} \\p_{41} & p_{42} & p_{43} & p_{44}\end{bmatrix} \cdot \begin{bmatrix}x_{1} \\x_{2} \\x_{3} \\x_{4}\end{bmatrix}} = {\quad\begin{bmatrix}{{p_{11}x_{1}} + {p_{12}x_{2}} + {p_{13}x_{3}} + {p_{14}x_{4}}} \\{{p_{21}x_{1}} + {p_{22}x_{2}} + {p_{23}x_{3}} + {p_{24}x_{4}}} \\{{p_{31}x_{1}} + {p_{32}x_{2}} + {p_{33}x_{3}} + {p_{34}x_{4}}} \\{{p_{41}x_{1}} + {p_{42}x_{2}} + {p_{43}x_{3}} + {p_{44}x_{4}}}\end{bmatrix}}}}}} & \left\lbrack {{Equation}\mspace{14mu} 22} \right\rbrack\end{matrix}$

In case of 4Tx-Rank 4, signals from four layers are multiplexed andtransmitted via respective antennas.

In one embodiment of the present invention, if a precoding matrix isconfigured in the form of an identity matrix, one antenna is able totransmit only a signal corresponding to a single layer. In this case,Equation 22 can be rewritten as the following equation 23.

$\begin{matrix}{y = {\begin{bmatrix}y_{1} \\y_{2} \\y_{3} \\y_{4}\end{bmatrix} = {{P \cdot x} = {{\begin{bmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{bmatrix} \cdot \begin{bmatrix}x_{1} \\x_{2} \\x_{3} \\x_{4}\end{bmatrix}} = \begin{bmatrix}x_{1} \\x_{2} \\x_{3} \\x_{4}\end{bmatrix}}}}} & \left\lbrack {{Equation}\mspace{14mu} 23} \right\rbrack\end{matrix}$

4Tx-Rank 4 Precoding Matrix (2)

In a 4Tx-Rank 4 codebook, if the number of Rank-4 precoding matrices isincreased, communication performance can also be increased. As thenumber of precoding matrices contained in a codebook increases, aprecoding matrix closer to an actual channel can be selected. Thus, thegreater the number of precoding matrices, the higher the performance.However, the selection of a precoding matrix in a codebook becomescomplicated, so that it is preferable that an appropriate number ofprecoding matrices should be included in such a codebook. However, incase of 4Tx-Rank 4, in order to transmit only a signal corresponding toa single layer via each antenna, the precoding matrix should be anidentity matrix, so that a signal corresponding to two or more layersshould sometimes be transmitted via a single antenna in case of usingseveral Rank 4 precoding matrices. Therefore, in order to minimize a CMvalue and increase the number of Rank 4 precoding matrices in acodebook, a specific element of the precoding matrix may be set to zero‘0’. In Equation 22, if it is assumed that two values of p_(k1), p_(k2),p_(k3) and p_(k4) in the signal (p_(k1)x₁+p_(k2)x₂+p_(k3)x₃+p_(k4)x₄)transmitted via each antenna are respectively set to zero ‘0’, a CM ofthe signal transmitted via each antenna can be maintained at a lowvalue.

In one embodiment of the present invention, the precoding matrix can beset to P₉ of the following equation 24, P₁₀ of the following equation25, or P₁₁ of the following equation 26.

$\begin{matrix}{\mspace{79mu} {{P_{9} = \begin{bmatrix}1 & A & 0 & 0 \\0 & 1 & B & 0 \\0 & 0 & 1 & C \\D & 0 & 0 & 1\end{bmatrix}},{1 \neq {A\; B\; C\; D}},A,B,C,{D \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2}}} \right\}}}} & \left\lbrack {{Equation}\mspace{14mu} 24} \right\rbrack \\{\mspace{79mu} {{P_{10} = {\begin{bmatrix}1 & 0 & 1 & 0 \\A & 0 & C & 0 \\0 & 1 & 0 & 1 \\0 & B & 0 & D\end{bmatrix}\mspace{14mu} {where}}},{A \neq C},{B \neq D},A,B,C,{D \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2}}} \right\}}}} & \left\lbrack {{Equation}\mspace{14mu} 25} \right\rbrack \\{\mspace{79mu} {{P_{11} = \begin{bmatrix}1 & 0 & 1 & 0 \\A & 0 & {- A} & 0 \\0 & 1 & 0 & 1 \\0 & B & 0 & {- B}\end{bmatrix}},A,{B \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2}}} \right\}}}} & \left\lbrack {{Equation}\mspace{14mu} 26} \right\rbrack\end{matrix}$

The precoding matrix P₉, P₁₀ or P₁₁ is an example of a precoding matrixfor transmitting a signal corresponding to a maximum of two layers viaeach antenna. As described above, the row/column permutation isperformed on the precoding matrix P₉, P₁₀ or P₁₁, so that signals ofdifferent layers can be transmitted via different antennas.

The precoding matrix P₁₁ is a unitary matrix, so that the advantages ofthe unitary precoding matrix can be utilized.

4Tx-Rank 4 Precoding Matrix (3)

In case of 4Tx-Rank 4, only one element among elements of each row of aprecoding matrix can be set to zero ‘0’. In case of using the abovemethod, a signal corresponding to three layers can be multiplexed andtransmitted via a single antenna, so that communication performance canbe improved. However, in the case of using the above-mentioned method, aCM value further increases, but the increased CM value may be lower thananother CM value acquired when all elements of the precoding matrix areeach set to any of other values except for zero ‘0’. Thus, theabove-mentioned method can be effectively utilized under a good SNRstatus in which a transmitter need not transmit data or information at amaximum transmission power.

In one embodiment of the present invention, the precoding matrix P canbe represented by P₁₂ of the following equation 27, P₁₃ of the followingequation 28, P₁₄ of the following equation 29, or P₁₅ of the followingequation 30.

$\begin{matrix}{\mspace{79mu} {{P_{12} = \begin{bmatrix}1 & m_{12} & m_{13} & 0 \\0 & 1 & m_{23} & m_{24} \\m_{31} & 0 & 1 & m_{34} \\m_{41} & m_{42} & 0 & 1\end{bmatrix}},{m_{ik} \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2}}} \right\}},\mspace{20mu} i,{k = 1},2,3,4}} & \left\lbrack {{Equation}\mspace{14mu} 27} \right\rbrack \\{\mspace{79mu} {{P_{13} = \begin{bmatrix}1 & 0 & 1 & 1 \\m_{21} & 0 & m_{23} & m_{24} \\0 & 1 & m_{33} & m_{34} \\0 & m_{42} & m_{43} & m_{44}\end{bmatrix}},{m_{ik} \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2}}} \right\}},\mspace{20mu} i,{k = 1},2,3,4}} & \left\lbrack {{Equation}\mspace{14mu} 28} \right\rbrack \\{\mspace{79mu} {{P_{14} = \begin{bmatrix}1 & 0 & 1 & 1 \\m_{21} & 0 & m_{23} & m_{24} \\m_{31} & 0 & m_{33} & m_{34} \\0 & 1 & m_{43} & m_{44}\end{bmatrix}},{m_{ik} \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2}}} \right\}},\mspace{20mu} i,{k = 1},2,3,4}} & \left\lbrack {{Equation}\mspace{14mu} 29} \right\rbrack \\{\mspace{79mu} {{P_{15} = \begin{bmatrix}1 & 1 & 1 & 0 \\0 & c & {- c} & c \\a & 0 & {- a} & {- a} \\b & {- b} & 0 & b\end{bmatrix}},a,b,{c \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2}}} \right\}}}} & \left\lbrack {{Equation}\mspace{14mu} 30} \right\rbrack\end{matrix}$

The precoding matrix P₁₅ illustrated in Equation 30 is a unitary matrix,so that the advantages of the unitary precoding matrix can be utilized.

A matrix acquired when a constant is multiplied by a specific column ofthe precoding matrix or another matrix acquired when row/columnpermutation is performed on the above-mentioned precoding matrix may beused as a part of a codebook.

Elements of the above-mentioned precoding matrices are selected from acomplex number that has an absolute value of 1 and a phase correspondingto any one of +0°, +45°, +90°, +135°, +180°, −135°, −90°, and −45°. Thatis, elements of the precoding matrix are selected from

$\left\{ {^{j\theta},{\theta \in \left\{ {0,\frac{\pi}{4},\frac{\pi}{2},\frac{3\pi}{4},\pi,\frac{5\pi}{4},\frac{6\pi}{4},\frac{7\pi}{4}} \right\}}} \right\}.$

For example, the above-mentioned selection has been disclosed only forillustrative purposes, and the elements of the precoding matrix may beselected from a set of complex numbers that have an absolute value of 1and different phases. For example, each element of the precoding matrixmay be selected from

$\left\{ {^{{j\theta} + a},{\theta \in \left\{ {0,\frac{\pi}{4},\frac{\pi}{2},\frac{3\pi}{4},\pi,\frac{5\pi}{4},\frac{6\pi}{4},\frac{7\pi}{4}} \right\}}} \right\}$

(where a is an arbitrary constant).

Power Balancing

In the meantime, transmission power balancing of respective antennasand/or transmission power balancing of respective layers may beconsidered to be an important matter in codebook design. If transmissionpowers of individual antennas are not adjusted for maximal uniformity,there arises a difference in performance between respective transmissionantennas. Likewise, if transmission powers of individual layers are notadjusted for maximal uniformity, there arises a difference inperformance between respective codewords.

Therefore, one embodiment of the present invention proposes a method fordesigning a precoding matrix in consideration of antenna power balancingusing norms of all elements (i.e., all elements of a specific row of theprecoding matrix) corresponding to individual antennas of the precodingmatrix. In more detail, the precoding matrix shown in the followingequation 31 may be utilized in the form of a antenna power balancingshown in the following equation 32.

$\begin{matrix}{\mspace{79mu} {P = {k \cdot \begin{bmatrix}p_{11} & \ldots & p_{1\; N_{L}} \\p_{21} & \ddots & p_{2\; N_{L}} \\\vdots & \ddots & \vdots \\p_{N_{T}1} & \ldots & p_{N_{T}N_{L}}\end{bmatrix}}}} & \left\lbrack {{Equation}\mspace{14mu} 31} \right\rbrack \\{P = {k \cdot \begin{bmatrix}\frac{p_{11}}{\sqrt{p_{11}^{2} + {\ldots \mspace{14mu} p_{1\; N_{L}}^{2}}}} & \ldots & \frac{p_{1\; N_{L}}}{\sqrt{p_{11}^{2} + {\ldots \mspace{14mu} p_{1\; N_{L}}^{2}}}} \\\frac{p_{21}}{\sqrt{p_{21}^{2} + {\ldots \mspace{14mu} p_{2\; N_{L}}^{2}}}} & \ddots & \frac{p_{2\; N_{L}}}{\sqrt{p_{21}^{2} + {\ldots \mspace{14mu} p_{2\; N_{L}}^{2}}}} \\\vdots & \ddots & \vdots \\\frac{p_{N_{T}1}}{\sqrt{p_{N_{T}1}^{2} + {\ldots \mspace{14mu} p_{N_{T}N_{L}}^{2}}}} & \ldots & \frac{p_{N_{T}N_{L}}}{\sqrt{p_{N_{T}1}^{2} + {\ldots \mspace{14mu} p_{N_{T}N_{L}}^{2}}}}\end{bmatrix}}} & \left\lbrack {{Equation}\mspace{14mu} 32} \right\rbrack\end{matrix}$

On the other hand, one embodiment of the present invention provides amethod for designing a precoding matrix in consideration of layer powerbalancing using norms of all elements (i.e., all elements of a specificcolumn of the precoding matrix) of individual layers. In more detail,the precoding matrix shown in the following equation 33 may be utilizedin the form of layer power balancing shown in the following equation 34.

$\begin{matrix}{\mspace{79mu} {P = {k \cdot \begin{bmatrix}p_{11} & \ldots & p_{1\; N_{L}} \\p_{21} & \ddots & p_{2\; N_{L}} \\\vdots & \ddots & \vdots \\p_{N_{T}1} & \ldots & p_{N_{T}N_{L}}\end{bmatrix}}}} & \left\lbrack {{Equation}\mspace{14mu} 33} \right\rbrack \\{P = {k \cdot \begin{bmatrix}\frac{p_{11}}{\sqrt{p_{11}^{2} + {\ldots \mspace{14mu} p_{N_{T}1}^{2}}}} & \ldots & \frac{p_{1\; N_{L}}}{\sqrt{p_{1\; N_{L}}^{2} + {\ldots \mspace{14mu} p_{N_{T}N_{L}}^{2}}}} \\\frac{p_{21}}{\sqrt{p_{11}^{2} + {\ldots \mspace{14mu} p_{N_{T}1}^{2}}}} & \ddots & \frac{p_{2\; N_{L}}}{\sqrt{p_{1\; N_{L}}^{2} + {\ldots \mspace{14mu} p_{N_{T}N_{L}}^{2}}}} \\\vdots & \ddots & \vdots \\\frac{p_{N_{T}1}}{\sqrt{p_{11}^{2} + {\ldots \mspace{14mu} p_{N_{T}1}^{2}}}} & \ldots & \frac{p_{N_{T}N_{L}}}{\sqrt{p_{1\; N_{L}}^{2} + {\ldots \mspace{14mu} p_{N_{T}N_{L}}^{2}}}}\end{bmatrix}}} & \left\lbrack {{Equation}\mspace{14mu} 34} \right\rbrack\end{matrix}$

In this case, differently from a Rank 2 precoding matrix, it can beinappropriate for the number of rows and the number of columns in a4Tx-Rank 3 precoding matrix to simultaneously perform the antenna powerbalancing and the power balancing. However, in a specific situation, forexample, in a system of using a layer shift that changes a layer usedfor transmission to another layer according to a specific pattern in atransmission mode, there occurs an effect in which a difference inperformance between layers is dispersed, the layer power balancing maybe relatively less important than the antenna power balancing.Therefore, one embodiment of the present invention proposes the use of aprecoding matrix acquired when the antenna power balancing is firstlycarried out on the condition that it is impossible to simultaneouslyperform the antenna power balancing and the layer power balancing.

In the meantime, the following precoding matrices among theabove-mentioned 4Tx-Rank 3 precoding matrices indicate that the antennapower balancing can be carried out because two symbols are transmittedto each layer, as denoted by the following equation 35.

$\begin{matrix}{{P_{0}^{\prime} = {k \cdot \begin{bmatrix}p_{11} & 0 & 0 \\0 & p_{22} & 0 \\0 & 0 & p_{33} \\\frac{p_{41}}{\sqrt{3}} & \frac{p_{42}}{\sqrt{3}} & \frac{p_{43}}{\sqrt{3}}\end{bmatrix}}}\mspace{14mu} {P_{4}^{\prime} = {k \cdot \begin{bmatrix}\frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}} \\\frac{X}{\sqrt{2}} & 0 & \frac{Z}{\sqrt{2}} \\0 & 1 & 0 \\0 & Y & 0\end{bmatrix}}}\mspace{14mu} {P_{5}^{\prime} = {k \cdot \begin{bmatrix}1 & 0 & 0 \\\frac{X}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\0 & \frac{Y}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\0 & 0 & Z\end{bmatrix}}}} & \left\lbrack {{Equation}\mspace{14mu} 35} \right\rbrack\end{matrix}$

Similarly, in case of the following precoding matrices among the4Tx-Rank 3 precoding matrices, because only one symbol is transmitted toone antenna, only the layer power balancing can be carried out as shownin the following equation 36.

$\begin{matrix}{{P_{1}^{\prime} = {{{k \cdot \begin{bmatrix}\frac{1}{\sqrt{2}} & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\\frac{X}{\sqrt{2}} & 0 & 0\end{bmatrix}}\mspace{14mu} P_{2}^{\prime}} = {k \cdot \begin{bmatrix}1 & 0 & 0 \\0 & \frac{1}{\sqrt{2}} & 0 \\0 & 0 & 1 \\0 & \frac{Y}{\sqrt{2}} & 0\end{bmatrix}}}}{P_{3}^{\prime} = {k \cdot \begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & \frac{1}{\sqrt{2}} \\0 & 0 & \frac{Z}{\sqrt{2}}\end{bmatrix}}}\mspace{14mu} {P_{7}^{\prime} = {k \cdot \begin{bmatrix}\frac{1}{\sqrt{3}} & 0 & \frac{Z}{\sqrt{3}} \\\frac{X}{\sqrt{3}} & \frac{1}{\sqrt{2}} & 0 \\0 & \frac{Y}{\sqrt{2}} & \frac{1}{\sqrt{3}} \\\frac{A}{\sqrt{3}} & 0 & \frac{C}{\sqrt{3}}\end{bmatrix}}}{P_{8}^{\prime} = {k \cdot \begin{bmatrix}\frac{1}{\sqrt{2}} & 0 & \frac{Z}{\sqrt{3}} \\\frac{X}{\sqrt{2}} & \frac{1}{\sqrt{3}} & 0 \\0 & \frac{Y}{\sqrt{3}} & \frac{1}{\sqrt{3}} \\0 & \frac{B}{\sqrt{3}} & \frac{C}{\sqrt{3}}\end{bmatrix}}}{P_{9}^{\prime} = {k \cdot \begin{bmatrix}\frac{1}{\sqrt{3}} & 0 & \frac{Z}{\sqrt{2}} \\\frac{X}{\sqrt{3}} & \frac{1}{\sqrt{3}} & 0 \\0 & \frac{Y}{\sqrt{3}} & \frac{1}{\sqrt{2}} \\\frac{A}{\sqrt{3}} & \frac{B}{\sqrt{3}} & 0\end{bmatrix}}}} & \left\lbrack {{Equation}\mspace{14mu} 36} \right\rbrack\end{matrix}$

In the meantime, in accordance with another embodiment of the presentinvention, from the viewpoint of simultaneous execution of both theantenna power balancing and the layer power balancing, the presentinvention proposes the 4Tx-Rank 3 precoding matrix including thefollowing precoding matrices denoted by Equation 37.

$\begin{matrix}{{P_{0}^{{(1)}^{\prime}} = {{\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\0 & 0 & 0\end{bmatrix}\mspace{14mu} P_{0}^{{(2)}^{\prime}}} = \begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 0 \\0 & 0 & 1\end{bmatrix}}}{P_{0}^{{(3)}^{\prime}} = {{\begin{bmatrix}1 & 0 & 0 \\0 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}\mspace{14mu} P_{0}^{{(4)}^{\prime}}} = \begin{bmatrix}0 & 0 & 0 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}}}} & \left\lbrack {{Equation}\mspace{14mu} 37} \right\rbrack\end{matrix}$

In other words, Equation 37 shows precoding matrices used as the4Tx-Rank 3 precoding matrices, and each precoding matrix of Equation 37is established to transmit no signal to a single specific antenna.

In the meantime, examples of the precoding matrix acquired when thelayer power balancing is carried out on the 4Tx-Rank 4 precoding matrixcan be represented by the following equation 38.

$\begin{matrix}{{P_{13}^{\prime} = {k \cdot \begin{bmatrix}\frac{1}{\sqrt{2}} & 0 & \frac{1}{2} & \frac{1}{2} \\\frac{m_{21}}{\sqrt{2}} & 0 & \frac{m_{23}}{2} & \frac{m_{24}}{2} \\0 & \frac{1}{\sqrt{2}} & \frac{m_{33}}{2} & \frac{m_{34}}{2} \\0 & \frac{m_{42}}{\sqrt{2}} & \frac{m_{43}}{2} & \frac{m_{44}}{2}\end{bmatrix}}}{P_{14}^{\prime} = {k \cdot \begin{bmatrix}\frac{1}{\sqrt{3}} & 0 & \frac{1}{2} & \frac{1}{2} \\\frac{m_{21}}{\sqrt{3}} & 0 & \frac{m_{23}}{2} & \frac{m_{24}}{2} \\\frac{m_{31}}{\sqrt{3}} & 0 & \frac{m_{33}}{2} & \frac{m_{34}}{2} \\0 & 1 & \frac{m_{43}}{2} & \frac{m_{44}}{2}\end{bmatrix}}}} & \left\lbrack {{Equation}\mspace{14mu} 38} \right\rbrack\end{matrix}$

<Codebook Pruning>

In a 4Tx system, precoding matrices corresponding to Rank 1, Rank 2,Rank 3, and Rank 4 can be used as elements of a codebook used intransmission and reception ends. However, in the case of using allprecoding matrices, the size of a codebook is excessively increased, sothat it is necessary to reduce the number of precoding matricessimultaneously while maintaining performance at an appropriate level.Embodiments capable of reducing the number of precoding matrices willhereinafter be described in detail. Methods for restricting thefollowing precoding matrix can be independently or simultaneouslyutilized.

Codebook Element Alphabet Restriction

Each element of the above-mentioned precoding matrices is selected froma complex number that has an absolute value of 1 and a phasecorresponding to any one of +0°, +45°, +90°, +135°, +180°, −135°, −90°,and −45°.

In one embodiment of the present invention, in order to reduce thenumber of precoding matrices, each element of the precoding matrices maybe selected from a complex number that has an absolute value of 1 and aphase corresponding to any one of +0°, +90°, +180°, and −90°. That is,each element of the precoding matrix may be selected from {1, j, −1,−j}.

Otherwise, each element of the precoding matrix may be extracted from asubset composed of N alphabetical letters among 8 alphabets which arespaced apart from each other by an angle of 45′.

Restriction to Unitary Precoding Matrix

In the case where individual column vectors contained in a precodingmatrix have orthogonality therebetween, the precoding matrix may be aunitary matrix or a partially unitary matrix. If the precoding matrixhas the above-mentioned characteristics, an additional gain can beobtained.

Thus, in accordance with one embodiment of the present invention,unitary matrices or partially unitary matrices among all theaforementioned precoding matrices are collected so that a codebook canbe formed.

For example, the row/column permutation is carried out on the precodingmatrices shown in the following equation 39 and the precoding matricesshown in the following equation 40 so as to obtain a few matrices, andthe obtained matrices are combined, so that a codebook can be generated.

$\begin{matrix}{\mspace{79mu} {{P^{(1)} = \begin{bmatrix}1 \\a \\b \\c\end{bmatrix}},{P^{(2)} = \begin{bmatrix}1 & 0 \\a & 0 \\0 & 1 \\0 & b\end{bmatrix}},{P^{(3)} = \begin{bmatrix}1 & 0 & 1 \\a & 0 & {- a} \\0 & 1 & 0 \\0 & b & 0\end{bmatrix}},{P_{1}^{(4)} = \begin{bmatrix}1 & 0 & 1 & 0 \\a & 0 & {- a} & 0 \\0 & 1 & 0 & 1 \\0 & b & 0 & {- b}\end{bmatrix}},{P_{2}^{(4)} = \begin{bmatrix}1 & 1 & 1 & 0 \\0 & c & {- c} & c \\a & 0 & {- a} & {- a} \\b & {- b} & 0 & b\end{bmatrix}},\mspace{20mu} {where},a,b,{c \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2}}} \right\}}}} & \left\lbrack {{Equation}\mspace{14mu} 39} \right\rbrack\end{matrix}$

Restriction to Nested Structure

When constructing the precoding matrices of Rank 1, Rank 2, Rank 3 andRank 4, in the case where the precoding matrix of Rank 2 or Rank 3 canbe constructed with column vectors of the Rank 4 precoding matrix, theconstructed precoding matrix is called a precoding matrix with a nestedstructure. If a specific Rank 4 precoding matrix is used as a part of aprecoding codebook, the Rank 3 precoding matrix should be configuredwith column vectors of the specific Rank 4 precoding matrix, such thatthere occurs a limitation in the construction of the precoding matrix.Thus, the codebook size can be limited according to the aforementionednorm or standard.

In one embodiment of the present invention, the precoding matrix of Rank1, Rank 2, Rank 3, or Rank 4 may have a nested structure.

For example, a codebook can be constructed with a combination ofmatrices acquired by performing the row/column permutation on theprecoding matrices shown in the following equation 40.

$\begin{matrix}{\mspace{79mu} {{{P^{(1)} = \begin{bmatrix}1 \\a \\b \\c\end{bmatrix}},{P^{(2)} = \begin{bmatrix}1 & 0 \\a & 0 \\0 & 1 \\0 & b\end{bmatrix}},{P^{(3)} = \begin{bmatrix}1 & 0 & 1 \\a & 0 & {- a} \\0 & 1 & 0 \\0 & b & 0\end{bmatrix}},\mspace{20mu} {P_{1}^{(4)} = \begin{bmatrix}1 & 0 & 1 & 0 \\a & 0 & {- a} & 0 \\0 & 1 & 0 & 1 \\0 & b & 0 & {- b}\end{bmatrix}}}\mspace{20mu} {{where},a,b,{c \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2}}} \right\}}}}} & \left\lbrack {{Equation}\mspace{14mu} 40} \right\rbrack\end{matrix}$

In addition to the matrices shown in the above-mentioned equations,other applicable matrices may also exist. It can be easily understoodthat the applicable matrices can be obtained by performing the rowpermutation and/or the column permutation on the above-mentionedmatrices. In the present invention, because the precoding matrix haselements each having a value of 0, a certain antenna may not be mappedto a specific input stream. This operation may be recognized as anantenna selection function.

II. Detailed Format of Codebook

Hereinafter, in the case where a codebook is designed to satisfy theabove-mentioned codebook design rule, a method for deciding a precodingmatrix for each rank contained in the codebook in consideration of achordal distance will be described in detail.

FIG. 9 is a conceptual diagram illustrating a chordal distance.

A chordal distance is well known as one of norms (or standards) forcomparing performances of various codebook sets. Herein, the term“chordal” indicates a straight line between two points located at thecircumference. Therefore, given a two-dimensional (2D) case, a chordaldistance indicates a distance between two points located at thecircumference of a circle (e.g., a unit circle) as shown in FIG. 9.

There is a need for the 4Tx-codebook to consider a four-dimensionalchordal distance, so that the following equation 41 can be used as achordal distance for selecting a codebook set.

$\begin{matrix}{{d_{c}\left( {P,Q} \right)} = {\frac{1}{\sqrt{2}}{{{PP}^{H} - {QQ}^{H}}}_{F}}} & \left\lbrack {{Equation}\mspace{14mu} 41} \right\rbrack\end{matrix}$

In Equation 41, P is P=[v₁ v₂ θv_(N)], and Q is Q=[u₁ u₂ . . . u_(N)],where v_(i) and u_(i) (i=1, 2, . . . N, N=4 in the case of 4Tx antennas)are principal vectors of the matrices P and Q, respectively. Inaddition,

${A}_{F} = {\sqrt{\sum\limits_{i = 1}^{m}\; {\sum\limits_{j = 1}^{n}\; {a_{ij}}^{2}}} = \sqrt{{trace}\left( {AA}^{H} \right)}}$

is the Frobenius norm of the matrix. The above-mentioned chordaldistance can also be measured by the following equation 42.

$\begin{matrix}\begin{matrix}{{d_{c}\left( {P,Q} \right)} = {\frac{1}{\sqrt{2}}{{{PP}^{H} - {QQ}^{H}}}_{F}}} \\{= \sqrt{n - {{trace}\left( {{AA}^{H}{BB}^{H}} \right)}}}\end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 42} \right\rbrack\end{matrix}$

where A and B are orthonormal generation magtrices for P and Qrespectively

The above-mentioned codebook design for the 4Tx system based on fourtransmission antennas will be described using the above-mentionedchordal distance concept. For convenience of description and betterunderstanding of the present invention, a factor related to the powerbalancing will be omitted from the following expressions.

Rank 2

Firstly, it is assumed that the following codebooks of three groupscapable of maintaining good CM performance about the 4Tx-Rank 2 systemare used.

$\begin{matrix}{{{Group}\mspace{14mu} 1\left( {\begin{bmatrix}1 & 0 \\X & 0 \\0 & 1 \\0 & Y\end{bmatrix},\begin{bmatrix}1 & 0 \\X & 0 \\0 & 1 \\0 & {- Y}\end{bmatrix},\begin{bmatrix}1 & 0 \\{- X} & 0 \\0 & 1 \\0 & Y\end{bmatrix},\begin{bmatrix}1 & 0 \\{- X} & 0 \\0 & 1 \\0 & {- Y}\end{bmatrix}} \right)}\mspace{20mu} {X,{Y \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},} \right\}}}} & \left\lbrack {{Equation}\mspace{14mu} 43} \right\rbrack \\{{{Group}\mspace{14mu} 2\left( {\begin{bmatrix}1 & 0 \\0 & 1 \\X & 0 \\0 & Y\end{bmatrix},\begin{bmatrix}1 & 0 \\0 & 1 \\X & 0 \\0 & {- Y}\end{bmatrix},\begin{bmatrix}1 & 0 \\0 & 1 \\{- X} & 0 \\0 & Y\end{bmatrix},\begin{bmatrix}1 & 0 \\0 & 1 \\{- X} & 0 \\0 & {- Y}\end{bmatrix}} \right)}\mspace{20mu} {X,{Y \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},} \right\}}}} & \; \\{{{Group}\mspace{14mu} 3\left( {\begin{bmatrix}1 & 0 \\0 & 1 \\0 & Y \\X & 0\end{bmatrix},\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- Y} \\X & 0\end{bmatrix},\begin{bmatrix}1 & 0 \\0 & 1 \\0 & Y \\{- X} & 0\end{bmatrix},\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- Y} \\{- X} & 0\end{bmatrix}} \right)}\mspace{20mu} {X,{Y \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},} \right\}}}} & \;\end{matrix}$

While the number of precoding matrices satisfying the above-mentionedformats may be a considerably high number, it is preferable that acodebook for including a predetermined number of precoding matrices bedesigned according to a reasonable norm. The following descriptionproposes a method for limiting the number of precoding matrices for eachrank to a predetermined number or less using the following norms.

First Norm (Norm 1): Chordal distance

Second Norm (Norm 2): Reference indicating whether the precodingmatrices are uniformly selected from individual groups. If the number ofprecoding matrices/vectors in a codebook is not divided by the number ofgroups, the precoding matrices are most uniformly selected inconsideration of the first norm (Norm 1).

The above-mentioned norm can be equally applied not only to Rank 3 butalso to Rank 4 to be described later.

In more detail, one embodiment of the present invention proposes amethod for selecting the set of precoding matrices from a codebook abouta specific rank using the Norm 1. In a first step, a chordal distanceabout all precoding matrix pairs contained in a single codebook iscalculated using Equation 42. For example, if four codebook sets exist,four minimum chordal distance values can be represented by the followingexpression.

d _(c,min) ¹=1, d _(c,min) ²=0.56, d _(c,min) ³=0.71 and d _(c,min)⁴=1  [Expression]

In the above expression, the higher the value of d_(c,min) ^(i) (where iis a codebook set number), the higher the system performance. Thus, itis preferable that first and fourth codebooks go to a next selectionstep.

In a second step, in order to support a variety of wireless channelenvironments, the present invention proposes a method for most uniformlyselecting the precoding matrices for each group. For example, inaccordance with the proposed method of the present invention, if thereare three codebook groups and 16 precoding matrices are needed as theRank-2 codebook, 5 precoding matrices are selected from two groups, and6 precoding matrices are selected from the remaining one group. Forexample, in accordance with the proposed method of the presentinvention, 5 precoding matrices are selected from first two groups, and6 precoding matrices are selected from the last one group. Oneembodiment of the present invention may consider a method for limitingalphabets of each precoding matrix as described above, in which, forexample, an alphabet ‘X’ may be limited to X=1, j, −1, or −j. Thefollowing description illustrates exemplary 4Tx Rank-2 codebooks capableof being configured by the above steps.

TABLE 1 Rank-2 codebook set 1-1 $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & j \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- j} & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- j} & 0 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\{- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & j \\{- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix}\quad$ Rank-2 codebook set 2-1 $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\{- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- j} & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & j \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\{- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- j} & 0 \\0 & {- j}\end{bmatrix}\quad$ Rank-2 codebook set 3-1 $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & j \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- j} & 0 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\{- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & 1\end{bmatrix}\quad$ Rank-2 codebook set 4-1 $\begin{bmatrix}1 & 0 \\0 & 1 \\{- j} & 0 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & j \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\{- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & j\end{bmatrix}\quad$ Rank-2 codebook set 5-1 $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- j} & 0 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\{- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & {- j}\end{bmatrix}\quad$ Rank-2 codebook set 6-1 $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- j} & 0 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & j \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- j} & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\{- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & j \\j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\{- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & {- j}\end{bmatrix}\quad$ Rank-2 codebook set 7-1 $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & j \\j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\{- 1} & 0\end{bmatrix}\quad$ Rank-2 codebook set 8-1 $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- j} & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & j \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\{- j} & 0\end{bmatrix}\quad$ Rank-2 codebook set 9-1 $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & j \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\{- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & {- 1}\end{bmatrix}\quad$ Rank-2 codebook set 10-1 $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & j \\j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- j} & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\{- 1} & 0\end{bmatrix}\quad$ Rank-2 codebook set 11-1 $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 1 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\{- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\{- j} & 0\end{bmatrix}\quad$ Rank-2 codebook set 12-1 $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\{- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & j \\j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- j} & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- j}\end{bmatrix}\quad$

The above-mentioned codebooks shown in Table 1 are disclosed only forillustrative purposes, and row permutation and/or column permutation maybe applied to all or some of the precoding matrices.

If the 4Tx Rank-2 codebook includes 15 precoding matrices, one precodingmatrix may be removed from a group of selecting the largest number ofprecoding matrices among individual precoding matrix groups. Thefollowing description shows exemplary 4Tx Rank-2 codebooks configured bythe above-mentioned schemes.

TABLE 2 Rank-2 codebook set 1-2 $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & j \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- j} & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- j} & 0 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\{- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & j \\{- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix}\quad$ Rank-2 codebook set 2-2 $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\{- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- j} & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & j \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\{- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- j} & 0 \\0 & {- j}\end{bmatrix}\quad$ Rank-2 codebook set 3-2 $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & j \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- j} & 0 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\{- 1} & 0\end{bmatrix}\quad$ Rank-2 codebook set 4-2 $\begin{bmatrix}1 & 0 \\0 & 1 \\{- j} & 0 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & j \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\{- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & j\end{bmatrix}\quad$ Rank-2 codebook set 5-2 $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- j} & 0 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\{- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & {- j}\end{bmatrix}\quad$ Rank-2 codebook set 6-2 $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- j} & 0 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & j \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- j} & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\{- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & j \\j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\{- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & {- j}\end{bmatrix}\quad$ Rank-2 codebook set 7-2 $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & j \\j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\{- 1} & 0\end{bmatrix}\quad$ Rank-2 codebook set 8-2 $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- j} & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & j \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\{- j} & 0\end{bmatrix}\quad$ Rank-2 codebook set 9-2 $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & j \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\{- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & {- 1}\end{bmatrix}\quad$ Rank-2 codebook set 10-2 $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\{- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- j} & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\{- 1} & 0\end{bmatrix}\quad$ Rank-2 codebook set 11-2 $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\{- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\{- j} & 0\end{bmatrix}\quad$ Rank-2 codebook set 12-2 $\begin{bmatrix}1 & 0 \\0 & 1 \\{- 1} & 0 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- j} \\j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\{- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- j} & 0 \\0 & 1 \\0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & j \\j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & 1 \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\{- j} & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\{- 1} & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\1 & 0 \\0 & 1 \\0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\j & 0 \\0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\0 & {- 1} \\1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\0 & 1 \\1 & 0 \\0 & j\end{bmatrix},$ $\begin{bmatrix}1 & 0 \\j & 0 \\0 & 1 \\0 & {- j}\end{bmatrix}\quad$

The codebooks shown in Table 2 are also disclosed only for illustrativepurposes, the row permutation and/or column permutation may be carriedout on all or some of precoding matrices of the codebooks.

Rank 3 First Embodiment

In order to design the 4Tx Rank-3 codebook so as to maintain good CMproperties, it is assumed that the following three precoding matrixgroups are used. For convenience of description, a factor related topower balancing will be omitted herein.

$\begin{matrix}{{{{Group}\mspace{14mu} 1\begin{pmatrix}{\begin{bmatrix}1 & 0 & 1 \\X & 0 & {- X} \\0 & 1 & 0 \\0 & Y & 0\end{bmatrix},\begin{bmatrix}1 & 0 & 1 \\X & 0 & {- X} \\0 & 1 & 0 \\0 & {- Y} & 0\end{bmatrix},} \\{\begin{bmatrix}0 & 1 & 0 \\0 & X & 0 \\1 & 0 & 1 \\Y & 0 & {- Y}\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\0 & {- X} & 0 \\1 & 0 & 1 \\Y & 0 & {- Y}\end{bmatrix}}\end{pmatrix}}{X,{Y \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}}} \right\}}}}{{Group}\mspace{14mu} 2\begin{pmatrix}{\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\X & 0 & {- X} \\0 & Y & 0\end{bmatrix},\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\X & 0 & {- X} \\0 & {- Y} & 0\end{bmatrix},} \\{\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & X & 0 \\Y & 0 & {- Y}\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & {- X} & 0 \\Y & 0 & {- Y}\end{bmatrix}}\end{pmatrix}}{X,{Y \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}}} \right\}}}{{Group}\mspace{14mu} 3\begin{pmatrix}{\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & Y & 0 \\X & 0 & {- X}\end{bmatrix},\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & {- Y} & 0 \\X & 0 & {- X}\end{bmatrix},} \\{\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\Y & 0 & {- Y} \\0 & X & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\Y & 0 & {- Y} \\0 & {- X} & 0\end{bmatrix}}\end{pmatrix}}{X,{Y \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}}} \right\}}}} & \left\lbrack {{Equation}\mspace{14mu} 44} \right\rbrack\end{matrix}$

In case of Rank 3, the present invention proposes a method forconstructing a codebook according to the above-mentioned Norm 1 and Norm2 in the same manner as in Rank 2. In more detail, a chordal distanceabout all precoding matrix combinations available in a codebook iscalculated using Equation 42, and then a minimum number of sets eachhaving a maximum chordal distance can be selected. In addition, thepresent invention proposes a method for most uniformly selecting theprecoding matrix from each group (Group 1, 2, or 3). If the letterdenoted by a precoding matrix component of each group is restricted to(1, j, −1, −j), the following codebook capable of satisfying a minimumchordal distance d_(c), =0.707 can be obtained.

TABLE 3 Rank-3 codebook set 1-1 $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\j & 0 & {- j} \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & j & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & j & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & {- j} & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & 1 & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0 \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\j & 0 & {- j} \\0 & {- j} & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & 1 & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1} \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0 \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & j & 0 \\1 & 0 & 1 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & {- j} & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & {- j} & 0 \\1 & 0 & 1 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & {- 1} & 0 \\1 & 0 & 1 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\j & 0 & {- j} \\0 & j & 0\end{bmatrix}\quad$ Rank-3 codebook set 2-1 $\begin{bmatrix}0 & 1 & 0 \\0 & {- 1} & 0 \\1 & 0 & 1 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & j & 0 \\1 & 0 & 1 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\j & 0 & {- j} \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & {- 1} & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1} \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0 \\0 & {- j} & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\j & 0 & {- j} \\0 & {- j} & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & {- 1} & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & j & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0 \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\1 & 0 & {- 1} \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1} \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\j & 0 & {- j} \\0 & {- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0 \\0 & 1 & 0\end{bmatrix}\quad$ Rank-3 codebook set 3-1 $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\j & 0 & {- j} \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & 1 & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & {- j} & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & {- 1} & 0 \\1 & 0 & 1 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\1 & 0 & {- 1} \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & 1 & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\j & 0 & {- j} \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & j & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0 \\0 & {- j} & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & {- j} & 0 \\1 & 0 & 1 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & {- 1} & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & {- j} & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0 \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0 \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & {- 1} & 0 \\1 & 0 & {- 1}\end{bmatrix}\quad$ Rank-3 codebook set 4-1 $\begin{bmatrix}1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0 \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1} \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & 1 & 0 \\1 & 0 & 1 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & {- 1} & 0 \\1 & 0 & 1 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & {- j} & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & 1 & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0 \\0 & {- j} & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & {- 1} & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0 \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & {- 1} & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & j & 0 \\1 & 0 & 1 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1} \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & {- j} & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1} \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\j & 0 & {- j} \\0 & {- j} & 0\end{bmatrix}\quad$ Rank-3 codebook set 5-1 $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1} \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & {- j} & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & 1 & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0 \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & {- 1} & 0 \\1 & 0 & 1 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & {- j} & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0 \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & j & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & {- 1} & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1} \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\1 & 0 & {- 1} \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\j & 0 & {- j} \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & j & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & {- j} & 0 \\1 & 0 & 1 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\j & 0 & {- j} \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0 \\0 & {- j} & 0\end{bmatrix}\quad$ Rank-3 codebook set 6-1 $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\j & 0 & {- j} \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1} \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & {- 1} & 0 \\1 & 0 & 1 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & {- j} & 0 \\1 & 0 & 1 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\j & 0 & {- j} \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\j & 0 & {- j} \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & {- j} & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & j & 0 \\1 & 0 & 1 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & j & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & j & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0 \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & 1 & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0 \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\j & 0 & {- j} \\0 & {- 1} & 0\end{bmatrix}\quad$ Rank-3 codebook set 7-1 ${\begin{bmatrix}0 & 1 & 0 \\0 & {- 1} & 0 \\1 & 0 & 1 \\j & 0 & {- j}\end{bmatrix},}\;$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1} \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & j & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0 \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0 \\0 & {- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0 \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & 1 & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & j & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & {- j} & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & 1 & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & {- j} & 0 \\1 & 0 & 1 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & {- 1} & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\j & 0 & {- j} \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & {- j} & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\j & 0 & {- j} \\0 & {- j} & 0\end{bmatrix}\quad$ Rank-3 codebook set 8-1 $\begin{bmatrix}0 & 1 & 0 \\0 & 1 & 0 \\1 & 0 & 1 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\j & 0 & {- j} \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0 \\0 & {- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\j & 0 & {- j} \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & j & 0 \\1 & 0 & 1 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & {- 1} & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1} \\0 & {- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0 \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\1 & 0 & {- 1} \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & {- j} & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0 \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & {- 1} & 0 \\1 & 0 & 1 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1} \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & {- j} & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1} \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & j & 0 \\j & 0 & {- j}\end{bmatrix},$

It should be noted that the row permutation and/or the columnpermutation may be carried out on all or some of precoding matrices ofthe above codebooks shown in Table 3.

If only 15 precoding matrices are included in the Rank-3 codebook, oneprecoding matrix of a group for selecting the largest number ofprecoding matrices among individual groups is removed from the codebooksshown in Table 3, so that the removed result may be configured as shownin the following Table 4.

TABLE 4 Rank-3 codebook set 1-2 $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\j & 0 & {- j} \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & j & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & j & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & {- j} & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & 1 & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0 \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\j & 0 & {- j} \\0 & {- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1} \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0 \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & j & 0 \\1 & 0 & 1 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & {- j} & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & {- j} & 0 \\1 & 0 & 1 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & {- 1} & 0 \\1 & 0 & 1 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\j & 0 & {- j} \\0 & j & 0\end{bmatrix}\quad$ Rank-3 codebook set 2-2 $\begin{bmatrix}0 & 1 & 0 \\0 & {- 1} & 0 \\1 & 0 & 1 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & j & 0 \\1 & 0 & 1 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\j & 0 & {- j} \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & {- 1} & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1} \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0 \\0 & {- j} & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\j & 0 & {- j} \\0 & {- j} & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & {- 1} & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0 \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\1 & 0 & {- 1} \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1} \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\j & 0 & {- j} \\0 & {- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0 \\0 & 1 & 0\end{bmatrix},$ Rank-3 codebook set 3-2 $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\j & 0 & {- j} \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & {- j} & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & {- 1} & 0 \\1 & 0 & 1 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\1 & 0 & {- 1} \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & 1 & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\j & 0 & {- j} \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & j & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0 \\0 & {- j} & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & {- j} & 0 \\1 & 0 & 1 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & {- 1} & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & {- j} & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0 \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0 \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & {- 1} & 0 \\1 & 0 & {- 1}\end{bmatrix}\quad$ Rank-3 codebook set 4-2 $\begin{bmatrix}1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0 \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1} \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & 1 & 0 \\1 & 0 & 1 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & {- j} & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & 1 & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0 \\0 & {- j} & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & {- 1} & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0 \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & {- 1} & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & j & 0 \\1 & 0 & 1 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1} \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & {- j} & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1} \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\j & 0 & {- j} \\0 & {- j} & 0\end{bmatrix}\quad$ Rank-3 codebook set 5-2 $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1} \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & {- j} & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & 1 & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0 \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & {- 1} & 0 \\1 & 0 & 1 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & {- j} & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0 \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & j & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & {- 1} & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1} \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\1 & 0 & {- 1} \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & j & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & {- j} & 0 \\1 & 0 & 1 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\j & 0 & {- j} \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0 \\0 & {- j} & 0\end{bmatrix}\quad$ Rank-3 codebook set 6-2 $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\j & 0 & {- j} \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1} \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & {- 1} & 0 \\1 & 0 & 1 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & {- j} & 0 \\1 & 0 & 1 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\j & 0 & {- j} \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\j & 0 & {- j} \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & j & 0 \\1 & 0 & 1 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & j & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & j & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0 \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & 1 & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0 \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\j & 0 & {- j} \\0 & {- 1} & 0\end{bmatrix}\quad$ Rank-3 codebook set 7-2 $\begin{bmatrix}0 & 1 & 0 \\0 & {- 1} & 0 \\1 & 0 & 1 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1} \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & j & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0 \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0 \\0 & {- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0 \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & 1 & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & j & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & 1 & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & {- j} & 0 \\1 & 0 & 1 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & {- 1} & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\j & 0 & {- j} \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & {- j} & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\j & 0 & {- j} \\0 & {- j} & 0\end{bmatrix},$ Rank-3 codebook set 8-2 $\begin{bmatrix}0 & 1 & 0 \\0 & 1 & 0 \\1 & 0 & 1 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\j & 0 & {- j} \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\1 & 0 & {- 1} \\0 & 1 & 0 \\0 & {- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\j & 0 & {- j} \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\0 & j & 0 \\1 & 0 & 1 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & {- 1} & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1} \\0 & {- j} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0 \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\1 & 0 & {- 1} \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & {- j} & 0 \\j & 0 & {- j}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\j & 0 & {- j} \\0 & 1 & 0 \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1} \\0 & {- 1} & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & {- j} & 0 \\1 & 0 & {- 1}\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & {- 1} \\0 & j & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 0 \\0 & j & 0 \\j & 0 & {- j}\end{bmatrix}\quad$

It should be noted that the row permutation and/or the columnpermutation may be carried out on all or some of the above precodingmatrices shown in Table 4.

Rank 3 Second Embodiment

In one embodiment of the present invention, a method for constructing acodebook using 6 precoding matrix groups capable of maintaining good CMproperties will hereinafter be described. The six 4Tx Rank-3 precodingmatrix groups for maintaining good CM properties can be represented bythe following equation 45.

$\begin{matrix}{\mspace{20mu} {{{{Group}\mspace{14mu} {1\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\X & 0 & 0\end{bmatrix}}},{{Group}\mspace{14mu} {2\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\X & 0 & 0\end{bmatrix}}},\mspace{20mu} {{Group}\mspace{14mu} {3\begin{bmatrix}0 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & 0 \\X & 0 & 0\end{bmatrix}}},{{Group}\mspace{14mu} {4\begin{bmatrix}1 & 0 & 0 \\0 & 0 & 1 \\0 & 1 & 0 \\X & 0 & 0\end{bmatrix}}},\mspace{20mu} {{Group}\mspace{14mu} {5\begin{bmatrix}1 & 0 & 0 \\X & 0 & 0 \\0 & 0 & 1 \\0 & 1 & 0\end{bmatrix}}},{{Group}\mspace{14mu} {6\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\X & 0 & 0 \\0 & 0 & 1\end{bmatrix}}},\mspace{20mu} {where}}{X \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2\mspace{11mu}}}} \right\}}}} & \left\lbrack {{Equation}\mspace{14mu} 45} \right\rbrack\end{matrix}$

An example of the Rank-3 codebook including 24 precoding matrixes from 6groups shown in Equation 45 is shown in the following table 5. In orderto reduce complexity, in the example shown in Table 5, letters denotedby precoding matrix elements are restricted to 1, −j, −1, and −j.

TABLE 5 $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\j & 0 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\{- j} & 0 & 0\end{bmatrix}\quad$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\j & 0 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\{- j} & 0 & 0\end{bmatrix}\quad$ $\begin{bmatrix}0 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & 0 \\1 & 0 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & 0 \\{- 1} & 0 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & 0 \\j & 0 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & 0 \\{- j} & 0 & 0\end{bmatrix}\quad$ $\begin{bmatrix}1 & 0 & 0 \\0 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 0 \\0 & 0 & 1 \\0 & 1 & 0 \\{- 1} & 0 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 0 \\0 & 0 & 1 \\0 & 1 & 0 \\j & 0 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 0 \\0 & 0 & 1 \\0 & 1 & 0 \\{- j} & 0 & 0\end{bmatrix}\quad$ $\begin{bmatrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1 \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 0 \\j & 0 & 0 \\0 & 0 & 1 \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 0 \\{- j} & 0 & 0 \\0 & 0 & 1 \\0 & 1 & 0\end{bmatrix}\quad$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\j & 0 & 0 \\0 & 0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\{- j} & 0 & 0 \\0 & 0 & 1\end{bmatrix}\quad$

For another example, the present invention proposes a method forutilizing the remaining groups other than a fourth group (Group 4)generated by applying column permutation to a first group (Group 1)among all groups shown in Equation 45. Generally, if three columnvectors are represented by [c1, c2, c3], 5 column permutation matricessuch as [c1, c3, c2], [c2, c1, c3], [c2, c3, c1], [c3, c2, c1], and [c3,c1, c2] can be generated, thus 6 matrices can be achieved.

The reason why the specific vector permutation matrix is not used asdescribed above is that an encoded sequence is mapped to a specificcolumn vector (or a specific layer) of the precoding matrix. It isassumed that two independently-encoded codewords in the above-mentionedprecoding matrix groups are mapped to different layers as describedbelow.

(1) A first codeword is mapped to a first layer.

(2) A second codeword is evenly distributed and mapped to second andthird layers.

On the assumption that the above codeword-layer mapping is used, aspecific column permutation does not generate a difference in averageSINR between different codewords. For example, permutation from a columnvector [c1, c2, c3] to another column vector [c1, c3, c2] may indicatethat only a layer of a second codeword is swapped. In this way, theswapping between two layers to which the same second codeword is evenlydistributed and mapped does not cause a variation in performance. Forsystems utilizing SIC receivers, correct decoding of a codeword giventransmission of plurality of codewords leads to performanceenhancements. This is because once a codeword is correctly decoded. Sothe correctly decoded codeword information can be used to cancel outspatial layer interference. In the case that transmission power ofmultiple antennas is uniformly normalized, some column vectors of theprecoding matrix may have larger transmission power. In the case thereis no layer shifting/permutation between all transmission layers, aspecific layer corresponding to the column vector of the precodingmatrix which column vectors has larger transmission power may havebetter performance. In case there is no layer shifting/permutationacross all transmitted layers, in order to fully utilized SIC receiversthe first layer, which the first codeword is solely mapped to is mappedto the precoding matrix column vector which has larger transmissionpower, and the second codeword which is mapped to second and third layeris mapped to precoding vector columns which has relatively smallertransmission power compared to the first layer. In the case abovecodeword-layer mapping is used, Precoding matrices as shown in [Equation46] can be used to further enhance performance in case SuccessiveInterference Cancellation (SIC) receiver algorithm is used.

$\begin{matrix}{{{{Group}\mspace{14mu} {1\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\X & 0 & 0\end{bmatrix}}},{{Group}\mspace{14mu} {2\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\X & 0 & 0\end{bmatrix}}},{{Group}\mspace{14mu} {3\begin{bmatrix}0 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & 0 \\X & 0 & 0\end{bmatrix}}},{{Group}\mspace{14mu} {4\begin{bmatrix}1 & 0 & 0 \\X & 0 & 0 \\0 & 0 & 1 \\0 & 1 & 0\end{bmatrix}}},{{Group}\mspace{14mu} {5\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\X & 0 & 0 \\0 & 0 & 1\end{bmatrix}}},{where}}{X \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2\mspace{11mu}}}} \right\}}} & \left\lbrack {{Equation}\mspace{14mu} 46} \right\rbrack\end{matrix}$

The following codebooks are exemplary 4Tx Rank-3 codebooks, each ofwhich restricts letters contained in each of the above precodingmatrices groups to 1, j, −1, and −j, and includes 20 precoding matrices.

TABLE 6 $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\j & 0 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\{- j} & 0 & 0\end{bmatrix}\quad$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\j & 0 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\{- j} & 0 & 0\end{bmatrix}\quad$ $\begin{bmatrix}0 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & 0 \\1 & 0 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & 0 \\{- 1} & 0 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & 0 \\j & 0 & 0\end{bmatrix},$ $\begin{bmatrix}0 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & 0 \\{- j} & 0 & 0\end{bmatrix}\quad$ $\begin{bmatrix}1 & 0 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1 \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 0 \\j & 0 & 0 \\0 & 0 & 1 \\0 & 1 & 0\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 0 \\{- j} & 0 & 0 \\0 & 0 & 1 \\0 & 1 & 0\end{bmatrix}\quad$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\{- 1} & 0 & 0 \\0 & 0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\j & 0 & 0 \\0 & 0 & 1\end{bmatrix},$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\{- j} & 0 & 0 \\0 & 0 & 1\end{bmatrix}\quad$

In the meantime, in accordance with another embodiment of the presentinvention, the number of precoding matrices required for acquiringoptimum performance from a high rank is less than the number ofprecoding matrices required for acquiring optimum performance from a lowrank, so that the present invention can restrict the Rank-3 codebook tohave below 24 precoding matrices. In this case, the present inventionmay evenly select the precoding matrices from 6 precoding matrix groupsusing the Norm 2.

TABLE 7 $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix}\quad$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\{- 1} & 0 & 0\end{bmatrix}\quad$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\j & 0 & 0\end{bmatrix}\quad$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\{- j} & 0 & 0\end{bmatrix}\quad$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\0 & 1 & 0\end{bmatrix}\quad$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\0 & {- 1} & 0\end{bmatrix}\quad$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\0 & j & 0\end{bmatrix}\quad$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\0 & {- j} & 0\end{bmatrix}\quad$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\0 & 0 & 1\end{bmatrix}\quad$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\0 & 0 & {- 1}\end{bmatrix}\quad$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\0 & 0 & j\end{bmatrix}\quad$ $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\0 & 0 & {- j}\end{bmatrix}\quad$

As can be seen from the example of Table 7, if e^(−jθ) is multiplied bya specific column vector, column permutation in a precoding matrix hasno influence upon improvement of performance, so that the number ofprecoding matrices contained in a codebook is limited to 12. Meanwhile,in accordance with one embodiment of the present invention, antennapermutation may be carried out to obtain antenna selection gain. Thisantenna permutation may also be implemented by row permutation of aprecoding matrix contained in the above-mentioned codebook.

Rank 3 Third Embodiment

In the third embodiment of the present invention, it is assumed that thefollowing 6 precoding matrix groups are considered as precoding matricescapable of maintaining good CM performance.

$\begin{matrix}{{{{Group}\mspace{14mu} {1\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\X & 0 & 0\end{bmatrix}}},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\0 & X & 0\end{bmatrix},\begin{bmatrix}0 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & X\end{bmatrix}}{{{Group}\mspace{14mu} {2\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 1 \\X & 0 & 0\end{bmatrix}}},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\0 & X & 0\end{bmatrix},\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\0 & 0 & X\end{bmatrix}}{{{Group}\mspace{14mu} {3\begin{bmatrix}0 & 0 & 1 \\0 & 1 & 0 \\1 & 0 & 0 \\X & 0 & 0\end{bmatrix}}},\begin{bmatrix}0 & 0 & 1 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & X & 0\end{bmatrix},\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\0 & 0 & X\end{bmatrix}}{{{Group}\mspace{14mu} {4\begin{bmatrix}0 & 1 & 0 \\0 & 0 & 1 \\1 & 0 & 0 \\0 & X & 0\end{bmatrix}}},\begin{bmatrix}0 & 0 & 1 \\1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & X\end{bmatrix}}{{{Group}\mspace{14mu} {5\begin{bmatrix}1 & 0 & 0 \\X & 0 & 0 \\0 & 0 & 1 \\0 & 1 & 0\end{bmatrix}}},\begin{bmatrix}0 & 1 & 0 \\0 & X & 0 \\0 & 0 & 1 \\1 & 0 & 0\end{bmatrix},\begin{bmatrix}0 & 0 & 1 \\0 & 0 & X \\1 & 0 & 0 \\0 & 1 & 0\end{bmatrix}}{{{Group}\mspace{14mu} {6\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\X & 0 & 0 \\0 & 0 & 1\end{bmatrix}}},\begin{bmatrix}0 & 1 & 0 \\1 & 0 & 0 \\0 & X & 0 \\0 & 0 & 1\end{bmatrix},\begin{bmatrix}0 & 0 & 1 \\0 & 1 & 0 \\0 & 0 & X \\1 & 0 & 0\end{bmatrix}}{where}{X \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2\mspace{11mu}}}} \right\}}} & \left\lbrack {{Equation}\mspace{14mu} 47} \right\rbrack\end{matrix}$

In case of a first group (Group 1) in Equation 47, it can be recognizedthat three permutation matrices are selected from [c1, c3, c2], [c2, c1,c3], [c2, c3, c1], [c3, c2, c1], and [c3, c1, c2]. In case of a fourthgroup (Group 4), it can be recognized that one constituent precodingmatrix is excluded, because the excluded precoding matrix has alreadybeen included in a first group (Group 1). It is preferable that thethird embodiment be utilized when a layer shift operation is not carriedout. The third embodiment can implement a layer shift using a codebookincluding a precoding matrix set upon which column permutation iscarried out. Thus, an information sequence may be mapped to all layers,so that an SINR difference between layers can be normalized.

The third embodiment can select a precoding matrix using the first norm(Norm 1) and the second norm (Norm 2).

Rank 3 Fourth Embodiment

The fourth embodiment considers the following three groups as precodingmatrix groups for maintaining good CM properties.

$\begin{matrix}{{{{G\; 1} = \begin{bmatrix}1 & 0 & a \\X & 0 & b \\0 & 1 & c \\0 & Y & d\end{bmatrix}},{{G\; 2} = \begin{bmatrix}1 & 0 & a^{\prime} \\0 & 1 & b^{\prime} \\X & 0 & c^{\prime} \\0 & Y & d^{\prime}\end{bmatrix}},{{G\; 3} = \begin{bmatrix}1 & 0 & a^{''} \\0 & 1 & b^{''} \\0 & Y & c^{''} \\X & 0 & d^{''}\end{bmatrix}}}\mspace{20mu} {{where}X,{Y \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},{\frac{{- 1} - j}{\sqrt{2}} - j},\frac{{- 1} + j}{\sqrt{2}}} \right\}}}} & \left\lbrack {{Equation}\mspace{14mu} 48} \right\rbrack\end{matrix}$

The last vectors

$\begin{bmatrix}a \\b \\c \\d\end{bmatrix},\begin{bmatrix}a^{\prime} \\b^{\prime} \\c^{\prime} \\d^{\prime}\end{bmatrix},\begin{bmatrix}a^{''} \\b^{''} \\c^{''} \\d^{''}\end{bmatrix}$

among precoding matrix groups shown in Equation 48 may be differentprecoding matrices such as DFT-based precoding vectors/matrices orhousehold-based precoding vectors/matrices. For example, an example ofthe last vectors may be a Rank-1 codebook of the 3GPP LTE system(Release 8 system). Preferably, in order to maintain orthogonal/partialunitary characteristics of the matrix

$\quad{\begin{bmatrix}1 & 0 & a \\X & 0 & b \\0 & 1 & c \\0 & Y & d\end{bmatrix},}$

the matrices

$\begin{bmatrix}1 & a \\X & b\end{bmatrix}\mspace{14mu} {{and}\mspace{14mu}\begin{bmatrix}1 & c \\Y & d\end{bmatrix}}$

must satisfy unitary characteristics. Similarly, matrices

$\begin{bmatrix}1 & a^{\prime} \\X & c^{\prime}\end{bmatrix}\mspace{14mu} {{and}\mspace{14mu}\begin{bmatrix}1 & b^{\prime} \\Y & d^{\prime}\end{bmatrix}}$

of the matrix

$\quad\begin{bmatrix}1 & 0 & a^{\prime} \\0 & 1 & b^{\prime} \\X & 0 & c^{\prime} \\0 & Y & d^{\prime}\end{bmatrix}$

and matrices

$\begin{bmatrix}1 & a^{''} \\X & d^{''}\end{bmatrix}\mspace{14mu} {{and}\mspace{11mu}\begin{bmatrix}1 & b^{''} \\Y & c^{''}\end{bmatrix}}$

of the matrix

$\quad\begin{bmatrix}1 & 0 & a^{''} \\0 & 1 & b^{''} \\0 & Y & c^{''} \\X & 0 & d^{''}\end{bmatrix}$

must satisfy unitary characteristics. This means that parameters mustsatisfy the following relationship.

In Group 1: a=1, b=−X, and c=−d·Y*

In Group 2: a′=1, b′=−X, and c′=−d′·Y*

In Group 3: a″=1, b″=−X, and c″=−d″·Y*  [Equation 49]

In this case, although a certain complex constant is multiplied by eachcolumn vector of a specific precoding matrix, this means that themultiplied results indicate the same precoding matrix, so that it isassumed that a, a′, or a″ is set to 1.

Preferably, the fourth embodiment may be applied to a case when layerpermutation is executed. The layer permutation operation indicates thata specific information sequence is cyclically mapped and transmitted toall layers so that SINR performance differences of individual layers arenormalized. If the same power is used in different layers, a datasequence of the last layer corresponding to the last column having novalue of 0 has the highest power from the viewpoint of a precodingoutput signal.

In case layer permutation is not used and enhanced SIC receiveralgorithm is used, layer which the first codeword is mapped to shouldpreferably correspond to the precoding vector column which thetransmission power is relatively larger than other precoding vectorcolumns. In case of [Equation 48] the third column may have largertransmission power than others. For cases which the first column ismapped to the first layer, the second column is mapped to the secondlayer, and the third column is mapped to the third layer, [Equation 48a]may be used instead of [Equation 48]. This precoding matrix structurewill allow enhanced performance in case no layer permutation is used andSIC receiver is used, due to the increased correct decoding probabilityof a entire codeword given a plurality of codeword transmission.

$\begin{matrix}{\mspace{20mu} {{{{G\; 1} = \begin{bmatrix}a & 0 & 1 \\b & 0 & X \\c & 1 & 0 \\d & Y & 0\end{bmatrix}},{{G\; 2} = \begin{bmatrix}a^{\prime} & 0 & 1 \\b^{\prime} & 1 & 0 \\c^{\prime} & 0 & X \\d^{\prime} & Y & 0\end{bmatrix}},\mspace{20mu} {{G\; 3} = \begin{bmatrix}a^{''} & 0 & 1 \\b^{''} & 1 & 0 \\c^{''} & 0 & X \\d^{''} & Y & 0\end{bmatrix}}}\mspace{20mu} {where}{X,{Y \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2\;}}} \right\}}}}} & \left\lbrack {{Equation}\mspace{14mu} 48a} \right\rbrack\end{matrix}$

Rank 3 Fifth Embodiment

In the fifth embodiment, it is assumed that the following groups shownin Equation 50 are used as precoding matrix groups for maintaining goodCM performance.

$\begin{matrix}{ {\left\lbrack {{Equation}\mspace{14mu} 50} \right\rbrack {{G\; 1} = \left( {{\begin{bmatrix}1 & 0 & a \\X & 0 & b \\0 & 1 & c \\0 & Y & d\end{bmatrix}\begin{bmatrix}0 & 1 & a \\0 & X & b \\1 & 0 & c \\Y & 0 & d\end{bmatrix}}\begin{bmatrix}a & 0 & 1 \\b & 0 & X \\c & 1 & 0 \\d & Y & 0\end{bmatrix}} \right)}}} & \; \\{{G\; 2} = \left( {{\begin{bmatrix}1 & 0 & a^{\prime} \\0 & 1 & b^{\prime} \\X & 0 & c^{\prime} \\0 & Y & d^{\prime}\end{bmatrix}\begin{bmatrix}0 & 1 & a^{\prime} \\1 & 0 & b^{\prime} \\0 & X & c^{\prime} \\Y & 0 & d^{\prime}\end{bmatrix}}\begin{bmatrix}a^{\prime} & 0 & 1 \\b^{\prime} & 1 & 0 \\c^{\prime} & 0 & X \\d^{\prime} & Y & 0\end{bmatrix}} \right)} & \; \\{{{G\; 3} = \left( {{\begin{bmatrix}1 & 0 & a^{''} \\0 & 1 & b^{''} \\0 & Y & c^{''} \\X & 0 & d^{''}\end{bmatrix}\begin{bmatrix}0 & 1 & a^{''} \\1 & 0 & b^{''} \\Y & 0 & c^{''} \\0 & X & d^{''}\end{bmatrix}}\begin{bmatrix}a^{''} & 0 & 1 \\b^{''} & 1 & 0 \\c^{''} & Y & 0 \\d^{''} & 0 & X\end{bmatrix}} \right)}{where}X,{Y \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2}}} \right\}}} & \;\end{matrix}$

The precoding matrix groups shown in Equation 50 are composed of aplurality of precoding matrices acquired when row permutation or columnpermutation is carried out on the structure of the fourth embodiment.The column vectors

$\begin{bmatrix}a \\b \\c \\d\end{bmatrix},\begin{bmatrix}a^{\prime} \\b^{\prime} \\c^{\prime} \\d^{\prime}\end{bmatrix},\begin{bmatrix}a^{''} \\b^{''} \\c^{''} \\d^{''}\end{bmatrix}$

in the precoding matrix groups shown in Equation 50 may be differentprecoding matrices such as DFT-based precoding vectors/matrices orhousehold-based precoding vectors/matrices. For example, an example ofthe above column vectors may be a Rank-1 codebook of the 3GPP LTE system(Release 8 system).

Similar to the fourth embodiment, in the fifth embodiment, it ispreferable that precoding matrix vectors be orthogonal to each other andelements other than a first value of 0 in all column vectors of eachprecoding matrix group be set to 1.

A codebook according to the fifth embodiment includes a precoding matrixgenerated when column permutation is carried out on the precodingmatrices of the fourth embodiment. As described above, the precodingmatrix having a column vector [c1, c2, c3] may have 6 column permutationprecoding matrices such as [c1, c3, c2], [c2, c1, c3], [c2, c3, c1],[c3, c2, c1], [c3, c1, c2] and [c3, c1, c2].

The reason why a specific column permutation is not included is thatsecond and third column permutations of the precoding matrix in a systemin which a first codeword is mapped to a first layer and a secondcodeword is distributed and mapped to second and third layers do notcause a difference in performance.

Rank 3 Sixth Embodiment

A precoding matrix according to the sixth embodiment is configured in aformat acquired when row permutation is carried out on a precodingmatrix of the codebook shown in the fourth embodiment, because theprecoding matrix of the sixth embodiment can be acquired by antennaswitching.

The precoding matrix groups according to the sixth embodiment can berepresented by the following equation 51.

                                                         [Equation  51]$\begin{matrix}{{G\; 1} = \left( {{{{{{\begin{bmatrix}1 & 0 & a \\X & 0 & b \\0 & 1 & c \\0 & Y & d\end{bmatrix}\begin{bmatrix}X & 0 & b \\1 & 0 & a \\0 & 1 & c \\0 & Y & d\end{bmatrix}}\begin{bmatrix}0 & 1 & c \\X & 0 & b \\1 & 0 & a \\0 & Y & d\end{bmatrix}}\begin{bmatrix}0 & Y & d \\X & 0 & b \\0 & 1 & c \\1 & 0 & a\end{bmatrix}}\begin{bmatrix}1 & 0 & a \\0 & 1 & c \\X & 0 & b \\0 & Y & d\end{bmatrix}}\begin{bmatrix}1 & 0 & a \\0 & Y & d \\0 & 1 & c \\X & 0 & b\end{bmatrix}}\begin{bmatrix}1 & 0 & a \\X & 0 & b \\0 & Y & d \\0 & 1 & c\end{bmatrix}} \right)} \\{{G\; 2} = \left( {{{{{{\begin{bmatrix}1 & 0 & a^{\prime} \\0 & 1 & b^{\prime} \\X & 0 & c^{\prime} \\0 & Y & d^{\prime}\end{bmatrix}\begin{bmatrix}0 & 1 & b^{\prime} \\1 & 0 & a^{\prime} \\X & 0 & c^{\prime} \\0 & Y & d^{\prime}\end{bmatrix}}\begin{bmatrix}X & 0 & c^{\prime} \\0 & 1 & b^{\prime} \\1 & 0 & a^{\prime} \\0 & Y & d^{\prime}\end{bmatrix}}\begin{bmatrix}0 & Y & d^{\prime} \\0 & 1 & b^{\prime} \\X & 0 & c^{\prime} \\1 & 0 & a^{\prime}\end{bmatrix}}\begin{bmatrix}1 & 0 & a^{\prime} \\X & 0 & c^{\prime} \\0 & 1 & b^{\prime} \\0 & Y & d^{\prime}\end{bmatrix}}\begin{bmatrix}1 & 0 & a^{\prime} \\0 & Y & d^{\prime} \\X & 0 & c^{\prime} \\0 & 1 & b^{\prime}\end{bmatrix}}\begin{bmatrix}1 & 0 & a^{\prime} \\0 & 1 & b^{\prime} \\0 & Y & d^{\prime} \\X & 0 & c^{\prime}\end{bmatrix}} \right)} \\{{{G\; 3} = \left( {{{{{{\begin{bmatrix}1 & 0 & a^{''} \\0 & 1 & b^{''} \\0 & Y & c^{''} \\X & 0 & d^{''}\end{bmatrix}\begin{bmatrix}0 & 1 & b^{''} \\1 & 0 & a^{''} \\0 & Y & c^{''} \\X & 0 & d^{''}\end{bmatrix}}\begin{bmatrix}0 & Y & c^{''} \\0 & 1 & b^{''} \\1 & 0 & a^{''} \\X & 0 & d^{''}\end{bmatrix}}\begin{bmatrix}X & 0 & d^{''} \\0 & 1 & b^{''} \\0 & Y & c^{''} \\1 & 0 & a^{''}\end{bmatrix}}\begin{bmatrix}1 & 0 & a^{''} \\0 & Y & c^{''} \\0 & 1 & b^{''} \\X & 0 & d^{''}\end{bmatrix}}\begin{bmatrix}1 & 0 & a^{''} \\X & 0 & d^{''} \\0 & Y & c^{''} \\0 & 1 & b^{''}\end{bmatrix}}\begin{bmatrix}1 & 0 & a^{''} \\0 & 1 & b^{''} \\X & 0 & d^{''} \\0 & Y & c^{''}\end{bmatrix}} \right)}X,{Y \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2}}} \right\}}}\end{matrix}$

[Equation 51]

The column vectors

$\begin{bmatrix}a \\b \\c \\d\end{bmatrix},\begin{bmatrix}a^{\prime} \\b^{\prime} \\c^{\prime} \\d^{\prime}\end{bmatrix},\begin{bmatrix}a^{''} \\b^{''} \\c^{''} \\d^{''}\end{bmatrix}$

or their row permutation formats may be different precoding matricessuch as DFT-based precoding vectors/matrices or household-basedprecoding vectors/matrices. For example, an example of the above columnvectors may be a Rank-1 codebook of the 3GPP LTE system (Release 8system).

Similar to the fourth embodiment, in the sixth embodiment, it ispreferable that column vectors of the precoding matrix be orthogonal toeach other and elements a, a′, or a″ are set to 1. An example of thecodebook according to the sixth embodiment can be represented by thefollowing equation 52.

$\begin{matrix}{{{G\; 1^{\prime}} = {{{{{{\begin{bmatrix}1 & 0 & 1 \\X & 0 & {- X} \\0 & 1 & c \\0 & Y & d\end{bmatrix}\begin{bmatrix}X & 0 & {- X} \\1 & 0 & 1 \\0 & 1 & c \\0 & Y & d\end{bmatrix}}\begin{bmatrix}0 & 1 & c \\X & 0 & {- X} \\1 & 0 & 1 \\0 & Y & d\end{bmatrix}}\begin{bmatrix}0 & Y & d \\X & 0 & {- X} \\0 & 1 & c \\1 & 0 & 1\end{bmatrix}}\begin{bmatrix}1 & 0 & 1 \\0 & 1 & c \\X & 0 & {- X} \\0 & Y & d\end{bmatrix}}\mspace{70mu}\begin{bmatrix}1 & 0 & 1 \\0 & Y & d \\0 & 1 & c \\X & 0 & {- X}\end{bmatrix}}\begin{bmatrix}1 & 0 & 1 \\X & 0 & {- X} \\0 & Y & d \\0 & 1 & c\end{bmatrix}}}{{G\; 2^{\prime}} = {{{{{{\begin{bmatrix}1 & 0 & 1 \\0 & 1 & b^{\prime} \\X & 0 & {- X} \\0 & Y & d^{\prime}\end{bmatrix}\begin{bmatrix}0 & 1 & b^{\prime} \\1 & 0 & 1 \\X & 0 & {- X} \\0 & Y & d^{\prime}\end{bmatrix}}\begin{bmatrix}X & 0 & {- X} \\0 & 1 & b^{\prime} \\1 & 0 & 1 \\0 & Y & d^{\prime}\end{bmatrix}}\begin{bmatrix}0 & Y & d^{\prime} \\0 & 1 & b^{\prime} \\X & 0 & {- X} \\1 & 0 & 1\end{bmatrix}}\begin{bmatrix}1 & 0 & 1 \\X & 0 & {- X} \\0 & 1 & b^{\prime} \\0 & Y & d^{\prime}\end{bmatrix}}\mspace{70mu}\begin{bmatrix}1 & 0 & 1 \\0 & Y & d^{\prime} \\X & 0 & {- X} \\0 & 1 & b^{\prime}\end{bmatrix}}\begin{bmatrix}1 & 0 & 1 \\0 & 1 & b^{\prime} \\0 & Y & d^{\prime} \\X & 0 & {- X}\end{bmatrix}}}{{G\; 3^{\prime}} = {{{{{{\begin{bmatrix}1 & 0 & 1 \\0 & 1 & b^{''} \\0 & Y & c^{''} \\X & 0 & {- X}\end{bmatrix}\begin{bmatrix}0 & 1 & b^{''} \\1 & 0 & 1 \\0 & Y & c^{''} \\X & 0 & {- X}\end{bmatrix}}\begin{bmatrix}0 & Y & c^{''} \\0 & 1 & b^{''} \\1 & 0 & 1 \\X & 0 & {- X}\end{bmatrix}}\begin{bmatrix}X & 0 & {- X} \\0 & 1 & b^{''} \\0 & Y & c^{''} \\1 & 0 & 1\end{bmatrix}}\begin{bmatrix}1 & 0 & 1 \\0 & Y & c^{''} \\0 & 1 & b^{''} \\X & 0 & {- X}\end{bmatrix}}\mspace{70mu}\begin{bmatrix}1 & 0 & 1 \\X & 0 & {- X} \\0 & Y & c^{''} \\0 & 1 & b^{''}\end{bmatrix}}\begin{bmatrix}1 & 0 & 1 \\0 & 1 & b^{''} \\X & 0 & {- X} \\0 & Y & c^{''}\end{bmatrix}}}{where}X,{Y \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2}}} \right\}}} & \left\lbrack {{Equation}\mspace{14mu} 52} \right\rbrack\end{matrix}$

Rank 3 Seventh Embodiment

A codebook according to the seventh embodiment is configured in a formatof row permutation of the codebook shown in the fifth embodiment. Anexample of the codebook according to the seventh embodiment can berepresented by the following equation 53.

$\begin{matrix}\; & \left\lbrack {{Equation}\mspace{14mu} 53} \right\rbrack \\{{G\; 1} = \begin{pmatrix}{\begin{bmatrix}1 & 0 & a \\X & 0 & b \\0 & 1 & c \\0 & Y & d\end{bmatrix},\begin{bmatrix}X & 0 & b \\1 & 0 & a \\0 & 1 & c \\0 & Y & d\end{bmatrix},\begin{bmatrix}0 & 1 & c \\X & 0 & b \\1 & 0 & a \\0 & Y & d\end{bmatrix},\begin{bmatrix}0 & Y & d \\X & 0 & b \\0 & 1 & c \\1 & 0 & a\end{bmatrix},\begin{bmatrix}1 & 0 & a \\0 & 1 & c \\X & 0 & b \\0 & Y & d\end{bmatrix},\begin{bmatrix}1 & 0 & a \\0 & Y & d \\0 & 1 & c \\X & 0 & b\end{bmatrix},\begin{bmatrix}1 & 0 & a \\X & 0 & b \\0 & Y & d \\0 & 1 & c\end{bmatrix}} \\{\begin{bmatrix}0 & 1 & a \\0 & X & b \\1 & 0 & c \\Y & 0 & d\end{bmatrix},\begin{bmatrix}0 & X & b \\0 & 1 & a \\1 & 0 & c \\Y & 0 & d\end{bmatrix},\begin{bmatrix}1 & 0 & c \\0 & X & b \\0 & 1 & a \\Y & 0 & d\end{bmatrix},\begin{bmatrix}Y & 0 & d \\0 & X & b \\1 & 0 & c \\0 & 1 & a\end{bmatrix},\begin{bmatrix}0 & 1 & a \\1 & 0 & c \\0 & X & b \\Y & 0 & d\end{bmatrix},\begin{bmatrix}0 & 1 & a \\Y & 0 & d \\1 & 0 & c \\0 & X & b\end{bmatrix},\begin{bmatrix}0 & 1 & a \\0 & X & b \\Y & 0 & d \\1 & 0 & c\end{bmatrix}} \\{\begin{bmatrix}a & 0 & 1 \\b & 0 & X \\c & 1 & 0 \\d & Y & 0\end{bmatrix},\begin{bmatrix}b & 0 & X \\a & 0 & 1 \\c & 1 & 0 \\d & Y & 0\end{bmatrix},\begin{bmatrix}c & 1 & 0 \\b & 0 & X \\a & 0 & 1 \\d & Y & 0\end{bmatrix},\begin{bmatrix}d & Y & 0 \\b & 0 & X \\c & 1 & 0 \\a & 0 & 1\end{bmatrix},\begin{bmatrix}a & 0 & 1 \\c & 1 & 0 \\b & 0 & X \\d & Y & 0\end{bmatrix},\begin{bmatrix}a & 0 & 1 \\d & Y & 0 \\c & 1 & 0 \\b & 0 & X\end{bmatrix},\begin{bmatrix}a & 0 & 1 \\b & 0 & X \\d & Y & 0 \\c & 1 & 0\end{bmatrix}}\end{pmatrix}} & \; \\{{G\; 2} = \begin{pmatrix}{{{{{{\begin{bmatrix}1 & 0 & a^{\prime} \\0 & 1 & b^{\prime} \\X & 0 & c^{\prime} \\0 & Y & d^{\prime}\end{bmatrix}\begin{bmatrix}0 & 1 & b^{\prime} \\1 & 0 & a^{\prime} \\X & 0 & c^{\prime} \\0 & Y & d^{\prime}\end{bmatrix}}\begin{bmatrix}X & 0 & c^{\prime} \\0 & 1 & b^{\prime} \\1 & 0 & a^{\prime} \\0 & Y & d^{\prime}\end{bmatrix}}\begin{bmatrix}0 & Y & d^{\prime} \\0 & 1 & b^{\prime} \\X & 0 & c^{\prime} \\1 & 0 & a^{\prime}\end{bmatrix}}\begin{bmatrix}1 & 0 & a^{\prime} \\X & 0 & c^{\prime} \\0 & 1 & b^{\prime} \\0 & Y & d^{\prime}\end{bmatrix}}\begin{bmatrix}1 & 0 & a^{\prime} \\0 & Y & d^{\prime} \\X & 0 & c^{\prime} \\0 & 1 & b^{\prime}\end{bmatrix}}\begin{bmatrix}1 & 0 & a^{\prime} \\0 & 1 & b^{\prime} \\0 & Y & d^{\prime} \\X & 0 & c^{\prime}\end{bmatrix}} \\{{{{{{\begin{bmatrix}0 & 1 & a^{\prime} \\1 & 0 & b^{\prime} \\0 & X & c^{\prime} \\Y & 0 & d^{\prime}\end{bmatrix}\begin{bmatrix}1 & 0 & b^{\prime} \\0 & 1 & a^{\prime} \\0 & X & c^{\prime} \\Y & 0 & d^{\prime}\end{bmatrix}}\begin{bmatrix}0 & X & c^{\prime} \\1 & 0 & b^{\prime} \\0 & 1 & a^{\prime} \\Y & 0 & d^{\prime}\end{bmatrix}}\begin{bmatrix}Y & 0 & d^{\prime} \\1 & 0 & b^{\prime} \\0 & X & c^{\prime} \\0 & 1 & a^{\prime}\end{bmatrix}}\begin{bmatrix}0 & 1 & a^{\prime} \\0 & X & c^{\prime} \\1 & 0 & b^{\prime} \\Y & 0 & d^{\prime}\end{bmatrix}}\begin{bmatrix}0 & 1 & a^{\prime} \\Y & 0 & d^{\prime} \\0 & X & c^{\prime} \\1 & 0 & b^{\prime}\end{bmatrix}}\begin{bmatrix}0 & 1 & a^{\prime} \\1 & 0 & b^{\prime} \\Y & 0 & d^{\prime} \\0 & X & c^{\prime}\end{bmatrix}} \\{{{{{{\begin{bmatrix}a^{\prime} & 0 & 1 \\b^{\prime} & 1 & 0 \\c^{\prime} & 0 & X \\d^{\prime} & Y & 0\end{bmatrix}\begin{bmatrix}b^{\prime} & 1 & 0 \\a^{\prime} & 0 & 1 \\c^{\prime} & 0 & X \\d^{\prime} & Y & 0\end{bmatrix}}\begin{bmatrix}c^{\prime} & 0 & X \\b^{\prime} & 1 & 0 \\a^{\prime} & 0 & 1 \\d^{\prime} & Y & 0\end{bmatrix}}\begin{bmatrix}d^{\prime} & Y & 0 \\b^{\prime} & 1 & 0 \\c^{\prime} & 0 & X \\a^{\prime} & 0 & 1\end{bmatrix}}\begin{bmatrix}a^{\prime} & 0 & 1 \\c^{\prime} & 0 & X \\b^{\prime} & 1 & 0 \\d^{\prime} & Y & 0\end{bmatrix}}\begin{bmatrix}a^{\prime} & 0 & 1 \\d^{\prime} & Y & 0 \\c^{\prime} & 0 & X \\b^{\prime} & 1 & 0\end{bmatrix}}\begin{bmatrix}a^{\prime} & 0 & 1 \\b^{\prime} & 1 & 0 \\d^{\prime} & Y & 0 \\c^{\prime} & 0 & X\end{bmatrix}}\end{pmatrix}} & \; \\{{G\; 3} = \begin{pmatrix}{{{{{{\begin{bmatrix}1 & 0 & a^{''} \\0 & 1 & b^{''} \\0 & Y & c^{''} \\X & 0 & d^{''}\end{bmatrix}\begin{bmatrix}0 & 1 & b^{''} \\1 & 0 & a^{''} \\0 & Y & c^{''} \\X & 0 & d^{''}\end{bmatrix}}\begin{bmatrix}0 & Y & c^{''} \\0 & 1 & b^{''} \\1 & 0 & a^{''} \\X & 0 & d^{''}\end{bmatrix}}\begin{bmatrix}X & 0 & d^{''} \\0 & 1 & b^{''} \\0 & Y & c^{''} \\1 & 0 & a^{''}\end{bmatrix}}\begin{bmatrix}1 & 0 & a^{''} \\0 & Y & c^{''} \\0 & 1 & b^{''} \\X & 0 & d^{''}\end{bmatrix}}\begin{bmatrix}1 & 0 & a^{''} \\X & 0 & d^{''} \\0 & Y & c^{''} \\0 & 1 & b^{''}\end{bmatrix}}\begin{bmatrix}1 & 0 & a^{''} \\0 & 1 & b^{''} \\X & 0 & d^{''} \\0 & Y & c^{''}\end{bmatrix}} \\{{{{{{\begin{bmatrix}0 & 1 & a^{''} \\1 & 0 & b^{''} \\Y & 0 & c^{''} \\0 & X & d^{''}\end{bmatrix}\begin{bmatrix}1 & 0 & b^{''} \\0 & 1 & a^{''} \\Y & 0 & c^{''} \\0 & X & d^{''}\end{bmatrix}}\begin{bmatrix}Y & 0 & c^{''} \\1 & 0 & b^{''} \\0 & 1 & a^{''} \\0 & X & d^{''}\end{bmatrix}}\begin{bmatrix}0 & X & d^{''} \\1 & 0 & b^{''} \\Y & 0 & c^{''} \\0 & 1 & a^{''}\end{bmatrix}}\begin{bmatrix}0 & 1 & a^{''} \\Y & 0 & c^{''} \\1 & 0 & b^{''} \\0 & X & d^{''}\end{bmatrix}}\begin{bmatrix}0 & 1 & a^{''} \\0 & X & d^{''} \\Y & 0 & c^{''} \\1 & 0 & b^{''}\end{bmatrix}}\begin{bmatrix}0 & 1 & a^{''} \\1 & 0 & b^{''} \\0 & X & d^{''} \\Y & 0 & c^{''}\end{bmatrix}} \\{{{{{{\begin{bmatrix}a^{''} & 0 & 1 \\b^{''} & 1 & 0 \\c^{''} & Y & 0 \\d^{''} & 0 & X\end{bmatrix}\begin{bmatrix}b^{''} & 1 & 0 \\a^{''} & 0 & 1 \\c^{''} & Y & 0 \\d^{''} & 0 & X\end{bmatrix}}\begin{bmatrix}c^{''} & Y & 0 \\b^{''} & 1 & 0 \\a^{''} & 0 & 1 \\d^{''} & 0 & X\end{bmatrix}}\begin{bmatrix}d^{''} & 0 & X \\b^{''} & 1 & 0 \\c^{''} & Y & 0 \\a^{''} & 0 & 1\end{bmatrix}}\begin{bmatrix}a^{''} & 0 & 1 \\c^{''} & Y & 0 \\b^{''} & 1 & 0 \\d^{''} & 0 & X\end{bmatrix}}\begin{bmatrix}a^{''} & 0 & 1 \\d^{''} & 0 & X \\c^{''} & Y & 0 \\b^{''} & 1 & 0\end{bmatrix}}\begin{bmatrix}a^{''} & 0 & 1 \\b^{''} & 1 & 0 \\d^{''} & 0 & X \\c^{''} & Y & 0\end{bmatrix}}\end{pmatrix}} & \; \\{{where}{X,{Y \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2}}} \right\}}}} & \;\end{matrix}$

The column vectors

$\begin{bmatrix}a \\b \\c \\d\end{bmatrix},\begin{bmatrix}a^{\prime} \\b^{\prime} \\c^{\prime} \\d^{\prime}\end{bmatrix},\begin{bmatrix}a^{''} \\b^{''} \\c^{''} \\d^{''}\end{bmatrix}$

or their row permutation formats may be different precoding matricessuch as DFT-based precoding vectors/matrices or household-basedprecoding vectors/matrices. For example, an example of the above columnvectors may be a Rank-1 codebook of the 3GPP LTE system (Release 8system).

Similar to the fourth embodiment, in the seventh embodiment, it ispreferable that column vectors of the precoding matrix be orthogonal toeach other and elements a, a′, or a″ are set to 1. It is preferable thatthe codebook according to this embodiment be used when antennapermutation is not carried out, because the antenna permutation effectcan be achieved by the precoding matrix to which row permutation iscarried out when using the codebook of the seventh embodiment.

An example of the codebook according to the seventh embodiment can berepresented by the following equation 54.

$\begin{matrix}{{\mspace{1046mu} \left\lbrack {{Equation}\mspace{14mu} 54} \right\rbrack}{G\; 1} = \begin{pmatrix}{{{{{{\begin{bmatrix}1 & 0 & 1 \\X & 0 & {- X} \\0 & 1 & c \\0 & Y & d\end{bmatrix}\begin{bmatrix}X & 0 & {- X} \\1 & 0 & 1 \\0 & 1 & c \\0 & Y & d\end{bmatrix}}\begin{bmatrix}0 & 1 & c \\X & 0 & {- X} \\1 & 0 & 1 \\0 & Y & d\end{bmatrix}}\begin{bmatrix}0 & Y & d \\X & 0 & {- X} \\0 & 1 & c \\1 & 0 & 1\end{bmatrix}}\begin{bmatrix}1 & 0 & 1 \\0 & 1 & c \\X & 0 & {- X} \\0 & Y & d\end{bmatrix}}\;\begin{bmatrix}1 & 0 & 1 \\0 & Y & d \\0 & 1 & c \\X & 0 & {- X}\end{bmatrix}}\begin{bmatrix}1 & 0 & 1 \\X & 0 & {- X} \\0 & Y & d \\0 & 1 & c\end{bmatrix}} \\{{{{{{\begin{bmatrix}0 & 1 & 1 \\0 & X & {- X} \\1 & 0 & c \\Y & 0 & d\end{bmatrix}\begin{bmatrix}0 & X & {- X} \\0 & 1 & 1 \\1 & 0 & c \\Y & 0 & d\end{bmatrix}}\begin{bmatrix}1 & 0 & c \\0 & X & {- X} \\0 & 1 & 1 \\Y & 0 & d\end{bmatrix}}\begin{bmatrix}Y & 0 & d \\0 & X & {- X} \\1 & 0 & c \\0 & 1 & 1\end{bmatrix}}\begin{bmatrix}0 & 1 & 1 \\1 & 0 & c \\0 & X & {- X} \\Y & 0 & d\end{bmatrix}}\begin{bmatrix}0 & 1 & 1 \\Y & 0 & d \\1 & 0 & c \\0 & X & {- X}\end{bmatrix}}\begin{bmatrix}0 & 1 & 1 \\0 & X & {- X} \\Y & 0 & d \\1 & 0 & c\end{bmatrix}} \\{{{{{{\begin{bmatrix}1 & 0 & 1 \\{- X} & 0 & X \\c & 1 & 0 \\d & Y & 0\end{bmatrix}\begin{bmatrix}{- X} & 0 & X \\1 & 0 & 1 \\c & 1 & 0 \\d & Y & 0\end{bmatrix}}\begin{bmatrix}c & 1 & 0 \\{- X} & 0 & X \\1 & 0 & 1 \\d & Y & 0\end{bmatrix}}\begin{bmatrix}d & Y & 0 \\{- X} & 0 & X \\c & 1 & 0 \\1 & 0 & 1\end{bmatrix}}\begin{bmatrix}1 & 0 & 1 \\c & 1 & 0 \\{- X} & 0 & X \\d & Y & 0\end{bmatrix}}\begin{bmatrix}1 & 0 & 1 \\d & Y & 0 \\c & 1 & 0 \\{- X} & 0 & X\end{bmatrix}}\begin{bmatrix}1 & 0 & 1 \\{- X} & 0 & X \\d & Y & 0 \\c & 1 & 0\end{bmatrix}}\end{pmatrix}} & \; \\{{G\; 2} = \begin{pmatrix}{{{{{{\begin{bmatrix}1 & 0 & 1 \\0 & 1 & b^{\prime} \\X & 0 & {- X} \\0 & Y & d^{\prime}\end{bmatrix}\begin{bmatrix}0 & 1 & b^{\prime} \\1 & 0 & 1 \\X & 0 & {- X} \\0 & Y & d^{\prime}\end{bmatrix}}\begin{bmatrix}X & 0 & {- X} \\0 & 1 & b^{\prime} \\1 & 0 & 1 \\0 & Y & d^{\prime}\end{bmatrix}}\begin{bmatrix}0 & Y & d^{\prime} \\0 & 1 & b^{\prime} \\X & 0 & {- X} \\1 & 0 & 1\end{bmatrix}}\begin{bmatrix}1 & 0 & 1 \\X & 0 & {- X} \\0 & 1 & b^{\prime} \\0 & Y & d^{\prime}\end{bmatrix}}\begin{bmatrix}1 & 0 & 1 \\0 & Y & d^{\prime} \\X & 0 & {- X} \\0 & 1 & b^{\prime}\end{bmatrix}}\begin{bmatrix}1 & 0 & 1 \\0 & 1 & b^{\prime} \\0 & Y & d^{\prime} \\X & 0 & {- X}\end{bmatrix}} \\{{{{{{\begin{bmatrix}0 & 1 & 1 \\1 & 0 & b^{\prime} \\0 & X & {- X} \\Y & 0 & d^{\prime}\end{bmatrix}\begin{bmatrix}1 & 0 & b^{\prime} \\0 & 1 & 1 \\0 & X & {- X} \\Y & 0 & d^{\prime}\end{bmatrix}}\begin{bmatrix}0 & X & {- X} \\1 & 0 & b^{\prime} \\0 & 1 & 1 \\Y & 0 & d^{\prime}\end{bmatrix}}\begin{bmatrix}Y & 0 & d^{\prime} \\1 & 0 & b^{\prime} \\0 & X & {- X} \\0 & 1 & 1\end{bmatrix}}\begin{bmatrix}0 & 1 & 1 \\0 & X & {- X} \\1 & 0 & b^{\prime} \\Y & 0 & d^{\prime}\end{bmatrix}}\begin{bmatrix}0 & 1 & 1 \\Y & 0 & d^{\prime} \\0 & X & {- X} \\1 & 0 & b^{\prime}\end{bmatrix}}\begin{bmatrix}0 & 1 & 1 \\1 & 0 & b^{\prime} \\Y & 0 & d^{\prime} \\0 & X & {- X}\end{bmatrix}} \\{{{{{{\begin{bmatrix}1 & 0 & 1 \\b^{\prime} & 1 & 0 \\{- X} & 0 & X \\d^{\prime} & Y & 0\end{bmatrix}\begin{bmatrix}b^{\prime} & 1 & 0 \\1 & 0 & 1 \\{- X} & 0 & X \\d^{\prime} & Y & 0\end{bmatrix}}\begin{bmatrix}{- X} & 0 & X \\b^{\prime} & 1 & 0 \\1 & 0 & 1 \\d^{\prime} & Y & 0\end{bmatrix}}\begin{bmatrix}d^{\prime} & Y & 0 \\b^{\prime} & 1 & 0 \\{- X} & 0 & X \\1 & 0 & 1\end{bmatrix}}\begin{bmatrix}1 & 0 & 1 \\{- X} & 0 & X \\b^{\prime} & 1 & 0 \\d^{\prime} & Y & 0\end{bmatrix}}\begin{bmatrix}1 & 0 & 1 \\d^{\prime} & Y & 0 \\{- X} & 0 & X \\b^{\prime} & 1 & 0\end{bmatrix}}\begin{bmatrix}1 & 0 & 1 \\b^{\prime} & 1 & 0 \\d^{\prime} & Y & 0 \\{- X} & 0 & X\end{bmatrix}}\end{pmatrix}} & \; \\{{G\; 3} = \begin{pmatrix}{{{{{{\begin{bmatrix}1 & 0 & 1 \\0 & 1 & b^{''} \\0 & Y & c^{''} \\X & 0 & {- X}\end{bmatrix}\begin{bmatrix}0 & 1 & b^{''} \\1 & 0 & 1 \\0 & Y & c^{''} \\X & 0 & {- X}\end{bmatrix}}\begin{bmatrix}0 & Y & c^{''} \\0 & 1 & b^{''} \\1 & 0 & 1 \\X & 0 & {- X}\end{bmatrix}}\begin{bmatrix}X & 0 & {- X} \\0 & 1 & b^{''} \\0 & Y & c^{''} \\1 & 0 & 1\end{bmatrix}}\begin{bmatrix}1 & 0 & 1 \\0 & Y & c^{''} \\0 & 1 & b^{''} \\X & 0 & {- X}\end{bmatrix}}\begin{bmatrix}1 & 0 & 1 \\X & 0 & {- X} \\0 & Y & c^{''} \\0 & 1 & b^{''}\end{bmatrix}}\begin{bmatrix}1 & 0 & 1 \\0 & 1 & b^{''} \\X & 0 & {- X} \\0 & Y & c^{''}\end{bmatrix}} \\{{{{{{\begin{bmatrix}0 & 1 & 1 \\1 & 0 & b^{''} \\Y & 0 & c^{''} \\0 & X & {- X}\end{bmatrix}\begin{bmatrix}1 & 0 & b^{''} \\0 & 1 & 1 \\Y & 0 & c^{''} \\0 & X & {- X}\end{bmatrix}}\begin{bmatrix}Y & 0 & c^{''} \\1 & 0 & b^{''} \\0 & 1 & 1 \\0 & X & {- X}\end{bmatrix}}\begin{bmatrix}0 & X & d^{''} \\1 & 0 & b^{''} \\Y & 0 & c^{''} \\0 & 1 & 1\end{bmatrix}}\begin{bmatrix}0 & 1 & 1 \\Y & 0 & c^{''} \\1 & 0 & b^{''} \\0 & X & {- X}\end{bmatrix}}\begin{bmatrix}0 & 1 & 1 \\0 & X & {- X} \\Y & 0 & c^{''} \\1 & 0 & b^{''}\end{bmatrix}}\begin{bmatrix}0 & 1 & 1 \\1 & 0 & b^{''} \\0 & X & {- X} \\Y & 0 & c^{''}\end{bmatrix}} \\{{{{{{\begin{bmatrix}1 & 0 & 1 \\b^{''} & 1 & 0 \\c^{''} & Y & 0 \\{- X} & 0 & X\end{bmatrix}\begin{bmatrix}b^{''} & 1 & 0 \\1 & 0 & 1 \\c^{''} & Y & 0 \\{- X} & 0 & X\end{bmatrix}}\begin{bmatrix}c^{''} & Y & 0 \\b^{''} & 1 & 0 \\1 & 0 & 1 \\{- X} & 0 & X\end{bmatrix}}\begin{bmatrix}{- X} & 0 & X \\b^{''} & 1 & 0 \\c^{''} & Y & 0 \\1 & 0 & 1\end{bmatrix}}\begin{bmatrix}1 & 0 & 1 \\c^{''} & Y & 0 \\b^{''} & 1 & 0 \\{- X} & 0 & X\end{bmatrix}}\begin{bmatrix}1 & 0 & 1 \\{- X} & 0 & X \\c^{''} & Y & 0 \\b^{''} & 1 & 0\end{bmatrix}}\begin{bmatrix}1 & 0 & 1 \\b^{''} & 1 & 0 \\{- X} & 0 & X \\c^{''} & Y & 0\end{bmatrix}}\end{pmatrix}} & \; \\{\mspace{79mu} {{where}\mspace{79mu} {X,{Y \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2}}} \right\}}}}} & \;\end{matrix}$

Reference for Selecting Additional Precoding Matrix

In addition to the Norm 1 and the Norm 2, this embodiment is designed toconsider another norm. In this norm, elements denoted by letterscontained in each precoding matrix group are not selected from amongeight values, but are limited to 1, j, −1 and −j, thus reducing thenumber of precoding matrices contained in a codebook.

In accordance with this embodiment, a codebook set including 16precoding matrices is considered. For example, Rank 1 DFT vectors about4Tx antennas can be represented as follows.

N×N DFT matrix (or Fourier Matrix) F_(N) based on a given component,such as F_(N)=e^(−j·2π/N) normalized to 1/√{square root over (N)} can berepresented by the following equation 55.

$\begin{matrix}{F_{N} = \begin{bmatrix}1 & 1 & 1 & \ldots & 1 \\1 & F_{N}^{1} & F_{N}^{2} & \ldots & F_{N}^{N - 1} \\\vdots & \vdots & \vdots & \ldots & \vdots \\1 & F_{N}^{({N - 1})} & F_{N}^{2 \cdot {({N - 1})}} & \ldots & F_{N}^{{({N - 1})} \cdot {({N - 1})}}\end{bmatrix}} & \left\lbrack {{Equation}\mspace{14mu} 55} \right\rbrack\end{matrix}$

Rank 1 DFT vectors about the 4Tx antennas composed of 16 column vectorslocated at the first four rows of Equation 55.

TABLE 8 $\quad\begin{matrix}{{{{{{{\begin{bmatrix}1 \\1 \\1 \\1\end{bmatrix}\mspace{14mu}\begin{bmatrix}1 \\e^{{- j}\frac{1}{8}\pi} \\e^{{- j}\frac{1}{4}\pi} \\e^{{- j}\frac{3}{8}\pi}\end{bmatrix}}\mspace{14mu}\begin{bmatrix}1 \\e^{{- j}\frac{1}{4}\pi} \\e^{{- j}\frac{1}{2}\pi} \\e^{{- j}\frac{3}{4}\pi}\end{bmatrix}}\mspace{14mu}\begin{bmatrix}1 \\e^{{- j}\frac{3}{8}\pi} \\e^{{- j}\frac{3}{4}\pi} \\e^{{- j}\frac{9}{8}\pi}\end{bmatrix}}\mspace{14mu}\begin{bmatrix}1 \\e^{{- j}\frac{1}{2}\pi} \\e^{{- j}\; \pi} \\e^{{- j}\frac{3}{2}\pi}\end{bmatrix}}\mspace{14mu}\begin{bmatrix}1 \\e^{{- j}\frac{5}{8}\pi} \\e^{{- j}\frac{5}{4}\pi} \\e^{{- j}\frac{15}{8}\pi}\end{bmatrix}}\mspace{14mu}\begin{bmatrix}1 \\e^{{- j}\frac{3}{4}\pi} \\e^{{- j}\frac{3}{2}\pi} \\e^{{- j}\frac{9}{4}\pi}\end{bmatrix}}\mspace{14mu}\begin{bmatrix}1 \\e^{{- j}\frac{7}{8}\pi} \\e^{{- j}\frac{7}{4}\pi} \\e^{{- j}\frac{5}{8}\pi}\end{bmatrix}} \\{{{{{{\begin{bmatrix}1 \\e^{{- j}\; \pi} \\1 \\e^{{- j}\; \pi}\end{bmatrix}\mspace{14mu}\begin{bmatrix}1 \\e^{{- j}\frac{9}{8}\pi} \\e^{{- j}\frac{1}{4}\pi} \\e^{{- j}\frac{11}{8}\pi}\end{bmatrix}}\mspace{14mu}\begin{bmatrix}1 \\e^{{- j}\frac{5}{4}\pi} \\e^{{- j}\frac{5}{2}\pi} \\e^{{- j}\frac{7}{4}\pi}\end{bmatrix}}\mspace{14mu}\begin{bmatrix}1 \\e^{{- j}\frac{11}{8}\pi} \\e^{{- j}\frac{11}{4}\pi} \\e^{{- j}\frac{1}{8}\pi}\end{bmatrix}}\mspace{14mu}\begin{bmatrix}1 \\e^{{- j}\frac{3}{2}\pi} \\e^{{- j}\; \pi} \\e^{{- j}\frac{1}{2}\pi}\end{bmatrix}}\mspace{14mu}\begin{bmatrix}1 \\e^{{- j}\frac{13}{8}\pi} \\e^{{- j}\frac{5}{4}\pi} \\e^{{- j}\frac{7}{8}\pi}\end{bmatrix}}\mspace{14mu}\begin{bmatrix}1 \\e^{{- j}\frac{7}{4}\pi} \\e^{{- j}\frac{3}{2}\pi} \\e^{{- j}\frac{5}{4}\pi}\end{bmatrix}} \\\begin{bmatrix}1 \\e^{{- j}\frac{15}{8}\pi} \\e^{{- j}\frac{7}{4}\pi} \\e^{{- j}\frac{13}{8}\pi}\end{bmatrix}\end{matrix}$

Next, 4Tx Rank 1 house hold vector (HH vector) may be represented by thefollowing Table 9.

TABLE 9 $\quad\begin{matrix}{{{{{{{\begin{bmatrix}1 \\1 \\1 \\1\end{bmatrix}\mspace{14mu}\begin{bmatrix}1 \\{- j} \\{- 1} \\j\end{bmatrix}}\mspace{14mu}\begin{bmatrix}1 \\{- 1} \\1 \\{- 1}\end{bmatrix}}\mspace{14mu}\begin{bmatrix}1 \\j \\{- 1} \\{- j}\end{bmatrix}}\mspace{14mu}\begin{bmatrix}1 \\\frac{1 - j}{\sqrt{2}} \\{- j} \\\frac{{- 1} - j}{\sqrt{2}}\end{bmatrix}}\mspace{14mu}\begin{bmatrix}1 \\\frac{{- 1} - j}{\sqrt{2}} \\j \\\frac{1 - j}{\sqrt{2}}\end{bmatrix}}\mspace{14mu}\begin{bmatrix}1 \\\frac{{- 1} + j}{\sqrt{2}} \\{- j} \\\frac{1 + j}{\sqrt{2}}\end{bmatrix}}\mspace{14mu}\begin{bmatrix}1 \\\frac{1 + j}{\sqrt{2}} \\j \\\frac{{- 1} + j}{\sqrt{2}}\end{bmatrix}} \\{{{{{{{\begin{bmatrix}1 \\1 \\{- 1} \\{- 1}\end{bmatrix}\mspace{14mu}\begin{bmatrix}1 \\{- j} \\1 \\{- j}\end{bmatrix}}\mspace{14mu}\begin{bmatrix}1 \\{- 1} \\{- 1} \\1\end{bmatrix}}\mspace{14mu}\begin{bmatrix}1 \\j \\1 \\j\end{bmatrix}}\mspace{14mu}\begin{bmatrix}1 \\1 \\1 \\{- 1}\end{bmatrix}}\mspace{14mu}\begin{bmatrix}1 \\1 \\{- 1} \\1\end{bmatrix}}\mspace{14mu}\begin{bmatrix}1 \\{- 1} \\1 \\1\end{bmatrix}}\mspace{14mu}\begin{bmatrix}1 \\{- 1} \\{- 1} \\{- 1}\end{bmatrix}}\end{matrix}$

Codebook Size Restriction

At least one of the first to third norms (Norm 1, Norm 2 and Norm 3) maybe used to limit the number of precoding matrices contained in acodebook. In this embodiment, codebook size restriction for each rank,especially, size restriction in a Rank 1 codebook, will be described indetail.

Presently, a downlink 4Tx codebook for the 3GPP LTE system hasprescribed that respective ranks have the same number ofvectors/matrices (i.e., 16 vectors/matrices). However, it is well knownin the art that the number of precoding matrices required to acquireoptimum performance from a high rank is less than the number ofprecoding matrices required to acquire optimum performance from a lowrank. For this purpose, this embodiment of the present inventionproposes a new codebook format in which the number of precoding matricesof a low rank is higher than that of a high rank so that individualranks have different numbers of precoding matrices.

In the meantime, a mobile communication system can support a variety oftransmission modes. It is assumed that an X-th transmission mode iseffectively used for a UE located at a cell edge so that the UE cansupport a closed loop operation using a Rank 1 Precoding MatrixIndicator (PMI). In this case, a Rank 1 PMI vector may be selected fromthe Rank 1 precoding matrices contained in an overall codebook composedof a plurality of precoding matrices of all ranks supporting a Y-thtransmission mode such as an open loop MIMO or closed loop MIMO. In thiscase, it is assumed that the X-th transmission mode is different fromthe Y-th transmission mode. For the Y-th transmission mode, the size ofthe Rank 1 codebook need not be configured as a power of 2. In addition,although the Rank 1 codebook size is configured as a power of 2, onlythe codebook size can be increased without higher performanceimprovement. Thus, this embodiment proposes a method for rationallyrestricting the codebook size simultaneously while having appropriateperformance so that the codebook can be expressed with a smaller amountof feedback information.

Firstly, it is assumed that numbers of precoding matrices of individualranks supporting the Y-th transmission mode are set to A—Rank 1, B—Rank2, C—Rank 3, and D-Rank 4 (where D≦C≦B≦A). In this case, the size of anoverall codebook is equal to the sum of A, B, C, and D. In order tosupport the above codebook size, m-bit signaling for satisfying thefollowing condition shown in Equation 56 may be needed.

A+B+C+D≦2^(m)  [Equation 56]

If a UE is configured to use the X-th transmission mode, a UE is able touse Rank 1 PMI information. It is preferable that 2^(n) Rank 1 PMIs(where n<m) be newly defined to reduce the number of bits required forsignaling. A variety of methods (1), (2), (3), (4), (5) and (6) may beused to reduce the number of signaling bits.

(1) Method 1

If possible, an even-th index is selected.

(2) Method 2

If possible, an odd-th index is selected.

(3) Method 3

Initial 2^(n) indexes are selected.

(4) Method 4

Last 2^(n) indexes are selected.

(5) Method 5

Indexes are arbitrarily selected.

(6) Method 6

Construction is achieved by signaling.

For example, for the Y-th transmission mode, 33 precoding matrices maybe given for Rank 1, 15 precoding matrices may be given for Rank 2, 15precoding matrices may be given for Rank 3, and 4 precoding matrices maybe given for Rank 4.

In this case, a variety of methods (1), (2), (3), (4), (5) and (6) forconstructing the Rank 1 codebook used for indicating only 16 precodingmatrices can be used.

(1) Method 1

If possible, an even-th index is selected.

(2) Method 2

If possible, an odd-th index is selected.

(3) Method 3

Initial 16 indexes are selected.

(4) Method 4

Last 16 indexes are selected.

(5) Method 5

Indexes are arbitrarily selected.

(6) Method 6

Construction is achieved by signaling.

In the meantime, a variety of methods (1), (2), (3) and (4) forconstructing the Rank 1 codebook used for indicating only 32 precodingmatrices can be used.

(1) Method 1

Initial 32 indexes are selected.

(2) Method 2

Last 32 indexes are selected.

(3) Method 3

Indexes are arbitrarily selected.

(4) Method 4

Construction is achieved by signaling.

If 16 downlink Rank 1 vectors are contained in the Rank 1 codebookincluding 32 precoding matrices, the following restriction methods (I)and (II) can be used.

The restriction method (I) corresponds to a case for constructing the16-sized Rank 1 codebook, and a detailed description thereof willhereinafter be described in detail.

A) 16 downlink Rank 1 vectors are selected.

B) The 16-sized Rank 1 codebook is selected regardless of downlink Rank1 vectors.

(1) Initial 16 indexes are selected.

(2) Last 16 indexes are selected.

(3) Indexes are arbitrarily selected.

(4) Construction is achieved by signaling.

The other restriction method (II) corresponds to another case forconstructing the 32-sized Rank 1 codebook, and a detailed descriptionthereof will hereinafter be described in detail.

A) Selection of 16 downlink Rank 1 vectors+additional vectors.

(1) Initial 16 indexes are selected.

(2) Last 16 indexes are selected.

(3) Indexes are arbitrarily selected.

(4) Construction is achieved by signaling.

B) Selection of 32-sized Rank 1 codebook regardless of downlink Rank 1vectors.

(1) Initial 32 indexes are selected.

(2) Last 32 indexes are selected.

(3) Indexes are arbitrarily selected.

(4) Construction is achieved by signaling.

The number of codebooks for each rank can be effectively constructedaccording to the above-mentioned schemes.

III. Apparatus Configuration

Chapter III will hereinafter disclose an improved structure to becontained in a UE, wherein the improved structure can maintain good PAPRor CM properties simultaneously while applying the MIMO scheme to uplinksignal transmission.

FIG. 10 is a block diagram illustrating a general base station (BS) anda general user equipment (UE).

Referring to FIG. 10, a base station (BS) 10 includes a processor 11, amemory 12, and a Radio Frequency (RF) module 13. The RF module 13 isused as a transmission/reception module for receiving an uplink signaland transmitting a downlink signal. The processor 11 may controldownlink signal transmission using downlink signal transmissioninformation (for example, a specific precoding matrix contained in acodebook for downlink signal transmission) stored in the memory 12.Otherwise, as an inverse process of the precoding process, the processor11 may control a signal reception process by multiplying uplink signalreception information (e.g., an uplink signal) stored in the memory 12by a Hermitian matrix of the same precoding matrix as a precoding matrixused in the UE 20.

The UE 20 may include a processor 21, a memory 22, and an RF module 23used as a transmission/reception module for transmitting an uplinksignal and receiving a downlink signal. The processor 21 may controluplink signal transmission using uplink signal transmission information(for example, a specific precoding matrix contained in theabove-mentioned codebook for uplink signal transmission) stored in thememory 22. Otherwise, as an inverse process of the precoding process,the processor 21 may control a signal reception process by multiplyingdownlink signal reception information (e.g., a downlink signal) storedin the memory 22 by a Hermitian matrix of the same precoding matrix as aprecoding matrix used in the UE 20.

In the meantime, a detailed description about a processor of the UE 20(or the BS 10), particularly, a structure for transmitting a signalusing the SC-FDMA scheme, will hereinafter be described. A processor fortransmitting a signal based on the SC-FDMA scheme in the 3GPP LTE systemand a processor for transmitting a signal based on an OFDM scheme in the3GPP LTE system will hereinafter be described, and a processor forenabling a UE to transmit an uplink signal using the SC-FDMA scheme aswell as the MIMO scheme will then be described below.

FIGS. 11 and 12 illustrate an SC-FDMA scheme for transmitting an uplinksignal in the 3GPP LTE system and an OFDMA scheme for transmitting adownlink signal in the 3GPP LTE system.

Referring to FIG. 11, not only a UE for transmitting an uplink signalbut also a base station (BS) for transmitting a downlink signal includesa Serial-to-Parallel converter 401, a subcarrier mapper 403, an M-pointIDFT module 404, a Parallel-to-Serial converter 405, and the like.However, a UE for transmitting a signal using the SC-FDMA scheme furtherincludes an N-point DFT module 402, and compensates for a predeterminedpart of the IDFT processing influence of the M-point IDFT module 404 sothat a transmission signal can have single carrier characteristics.

FIG. 12 shows the relationship between a block diagram for an uplinksignal process prescribed in TS 36.211 including the 3GPP LTE systemspecification and a processor for transmitting a signal using theSC-FDMA scheme. In accordance with the TS 36.211, each UE scrambles atransmission signal using a specific scrambling sequence so as totransmit an uplink signal, and the scrambled signal is modulated so thatcomplex symbols are generated. After that, transform precoding forperforming a DFT spreading process on complex symbols is carried out.That is, a transform precoder prescribed in TS 36.211 may correspond toan N-point DFT module. Thereafter, the DFT-spread signal may be mappedto a specific resource element according to a resource block (RB)-basedmapping rule by a resource element mapper, and it can be recognized thatthis operation corresponds to the subcarrier mapper shown in FIG. 11.The signal mapped to the resource element is M-point IDFT orIFFT-processed by the SC-FDMA signal generator, parallel-to-serialconversion is performed on the IDFT or IFFT processed result, and then acyclic prefix (CP) is added to the P/S conversion result.

In the meantime, FIG. 12 further shows a processor of a base station(BS) that is used to receive a signal which has been received in thebase station through the above-mentioned processes.

In this way, the processor for SC-FDMA transmission in the 3GPP LTEsystem does not include a structure for utilizing the MIMO scheme.Therefore, the BS processor for MIMO transmission in the 3GPP LTE systemwill be described first, and a processor for transmitting an uplinksignal by combining the SC-FDMA scheme with the MIMO scheme using theabove BS processor will then be described.

FIG. 13 is a block diagram illustrating a processor for enabling thebase station (BS) to transmit a downlink signal using the MIMO scheme inthe 3GPP LTE system.

A base station (BS) in the 3GPP LTE system can transmit one or morecodewords via a downlink. Therefore, one or more codewords may beprocessed as complex symbols by the scrambling module 301 and themodulation mapper 302 in the same manner as in the uplink operationshown in FIG. 12. Thereafter, the complex symbols are mapped to aplurality of layers by the layer mapper 303, and each layer ismultiplied by a predetermined precoding matrix selected according to thechannel status and is then allocated to each transmission antenna by theprecoding module 304. The processed transmission signals of individualantennas are mapped to time-frequency resource elements to be used fordata transmission by the resource element mapper 305. Thereafter, themapped result may be transmitted via each antenna after passing throughthe OFDMA signal generator 306.

However, if a downlink signal scheme shown in FIG. 13 is used in the3GPP LTE system, PAPR or CM properties may be degraded. Thus, it isnecessary for a UE to effectively combine the SC-FDMA scheme formaintaining good PAPR and CM properties described in FIGS. 11 and 12with the MIMO scheme shown in FIG. 13, and a UE for performing precodingusing the precoding matrix capable of maintaining good PAPR and CMproperties described in the above embodiment must be constructed.

In accordance with one embodiment of the present invention, it isassumed that a UE for transmitting an uplink signal via multipleantennas (multi-antenna) includes multiple antennas (not shown) fortransmitting and receiving signals. Referring to FIG. 10, the UE 20includes a memory 22 for storing a codebook, and a processor 21 that areconnected to multiple antennas (not shown) and the memory 22 so as toprocess uplink signal transmission. In this case, the codebook stored inthe memory 22 includes precoding matrices established in a manner that asingle layer signal is transmitted to each of the multiple antennas. Theprocessor 21 of the UE configured as described above will hereinafter bedescribed in detail.

FIG. 14 illustrates a processor of the UE according to one embodiment ofthe present invention.

Referring to FIG. 14, the processor of the UE 20 according to oneembodiment of the present invention includes a codeword to layer mapper1401 for mapping uplink signals to a predetermined number of layerscorresponding to a specific rank, a predetermined number of DFT modules1402 for performing Discrete Fourier Transform (DFT) spreading on eachof the predetermined number of layer signals, and a precoder 1403. Theprecoder 1403 selects a specific precoding matrix established in amanner that one layer signal is transmitted to each antenna 1405 so asto precode a DFT-spread resultant layer signal received from the DFTmodule 1402. Particularly, in this embodiment of the present invention,each DFT module 1402 performs spreading of each layer signal, this DFTmodule 1402 for spreading each layer signal is located just before theprecoder 1403. When the precoder 1403 performs precoding, the precoder1403 is configured such that each layer signal is mapped to one antennaand then transmitted via the mapped antenna, so that single carriercharacteristics of each layer signal are maintained and good PAPR and CMproperties are also maintained. In the meantime, the UE 20 furtherincludes a transmission module. The transmission module performs aprocess constructing an SC-FDMA symbol upon the precoded signal, andtransmits the resultant precoded signal to the base station (BS) viamultiple antennas 1405.

In the meantime, the precoder 1403 selects a precoding matrix to be usedfor signal transmission from among the codebook stored in the memory 22,and performs precoding on the selected precoding matrix. Preferably,these precoding matrices may be precoding matrices established forequalizing transmission powers of multiple antennas and/or transmissionpowers of respective layers.

The number of multiple antennas 1405 may be 2 or 4. The processor of theUE according to one embodiment of the present invention may furtherperform not only a layer shift function for periodically oraperiodically changing a layer mapped to a specific codeword but also anantenna shift function for periodically or aperiodically changing anantenna via which a specific layer signal is transmitted. The layershift function may be performed by the layer mapper 1401 separately fromthe precoding of the precoder 1403, or may also be performed throughcolumn permutation of the precoding matrix when the precoder 1403performs precoding. In addition, the antenna shift function may also becarried out separately from the precoding of the precoder 1403, or mayalso be performed through row permutation of the precoding matrix.

The exemplary embodiments described hereinabove are combinations ofelements and features of the present invention. The elements or featuresmay be considered selective unless otherwise mentioned. Each element orfeature may be practiced without being combined with other elements orfeatures.

Further, the embodiments of the present invention may be constructed bycombining parts of the elements and/or features. Operation ordersdescribed in the embodiments of the present invention may be rearranged.Some constructions or characteristics of any one embodiment may beincluded in another embodiment and may be replaced with correspondingconstructions or characteristics of another embodiment. It is apparentthat the present invention may be embodied by a combination of claimswhich do not have an explicit cited relation in the appended claims ormay include new claims by amendment after application.

The embodiments of the present invention may be achieved by variousmeans, for example, hardware, firmware, software, or a combinationthereof. In a hardware configuration, the embodiments of the presentinvention may be implemented by one or more application specificintegrated circuits (ASICs), digital signal processors (DSPs), digitalsignal processing devices (DSPDs), programmable logic devices (PLDs),field programmable gate arrays (FPGAs), processors, controllers,microcontrollers, microprocessors, etc.

In a firmware or software configuration, the embodiments of the presentinvention may be achieved by a module, a procedure, a function, etc.performing the above-described functions or operations. Software codemay be stored in a memory unit and driven by a processor. The memoryunit may be located at the interior or exterior of the processor and maytransmit data to and receive data from the processor via various knownmeans.

It will be apparent to those skilled in the art that variousmodifications and variations can be made in the present inventionwithout departing from the spirit or scope of the invention. Therefore,the above-mentioned detailed description must be considered only forillustrative purposes instead of restrictive purposes. The scope of thepresent invention must be decided by a rational analysis of claims, andall modifications within equivalent ranges of the present invention arecontained in the scope of the present invention. It is apparent that thepresent invention may be embodied by a combination of claims which donot have an explicit cited relation in the appended claims or mayinclude new claims by amendment after application.

As apparent from the above description, the present invention canmaintain PAPR or CM properties while transmitting uplink signals using aMIMO scheme.

In addition, the present invention uniformly controls or adjustsantenna/layer transmission power, minimizes an amount of signalingoverhead required for precoding matrix information, and acquires amaximum diversity gain.

The present invention is applicable to a wideband wireless mobilecommunication system.

It will be apparent to those skilled in the art that variousmodifications and variations can be made in the present inventionwithout departing from the spirit or scope of the inventions. Thus, itis intended that the present invention covers the modifications andvariations of this invention provided they come within the scope of theappended claims and their equivalents.

What is claimed is:
 1. A method for controlling a user equipment (UE) totransmit uplink signals via 4 antennas, the method comprising: mappingthe uplink signals to 3 layers; performing Discrete Fourier Transform(DFT) spreading upon each of the 3 layers; precoding each of theDFT-spread layers by using a specific precoding matrix selected fromamong a prestored codebook for rank 3, wherein every precoding matrix inthe prestored codebook for rank 3 is established in a manner that eachone of the 4 antennas transmits only one layer of the 3 layers, and afirst column has 2 non-zero elements while each of a second and a thirdcolumns respectively has one non-zero element; and transmitting theprecoded signals to a base station (BS) via the 4 antennas by performinga predetermined process for constructing a Single Carrier-FrequencyDivision Multiple Access (SC-FDMA) symbol upon the precoded signals. 2.The method according to claim 1, wherein every precoding matrix in theprestored codebook for rank 3 is established in a manner that the 4antennas have uniform transmission power therebetween.
 3. The methodaccording to claim 1, wherein the uplink signals comprise 2 codewords,and wherein a first codeword of the 2 codewords is mapped to a firstlayer of the 3 layers while a second codeword of the 2 codewords isalternately mapped to a second and a third layers of the 3 layers. 4.The method according to claim 1, wherein the prestored codebook for rank3 includes 6 types of precoding matrixes, wherein one of the 6 types ofprecoding matrix is configured in a form of $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\X & 0 & 0\end{bmatrix},$ and satisfying a condition of${X \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2}}} \right\}},$and wherein individual rows of the precoding matrix respectivelycorrespond to four antennas of the 4 antennas, and individual columns ofthe precoding matrix respectively correspond to layers.
 5. A userequipment (UE) for transmitting uplink signals, the UE comprising: 4antennas for transmitting and receiving signals; a memory for storing acodebook for rank 3, wherein every precoding matrix in the prestoredcodebook for rank 3 is established in a manner that each one of the 4antennas transmits only one layer of 3 layers, and a first column has 2non-zero elements while each of a second and a third columnsrespectively has one non-zero element; and a processor connected to the4 antennas and the memory so as to process transmission of the uplinksignals, wherein the processor includes: a layer mapper for mapping theuplink signals to the 3 layers; a Discrete Fourier Transform (DFT)module for performing DFT spreading upon each of the 3 layers; aprecoder for precoding each of the DFT-spread layer signals receivedfrom the DFT module by selecting a specific precoding matrix from amongthe codebook for rank 3 stored in the memory; and a transmission modulefor performing a predetermined process for constructing a SingleCarrier-Frequency Division Multiple Access (SC-FDMA) symbol upon theprecoded signals, and transmitting the processed signals to a basestation (BS) via the 4 antennas.
 6. The UE according to claim 5, whereinevery precoding matrix in the codebook for rank 3 is established in amanner that the 4 antennas have uniform transmission power therebetween.7. The UE according to claim 5, wherein the uplink signals comprise 2codewords, and wherein a first codeword of the 2 codewords is mapped toa first layer of the 3 layers while a second codeword of the 2 codewordsis alternately mapped to a second and a third layers of the 3 layers. 8.The UE according to claim 5, wherein the prestored codebook for rank 3includes 6 types of precoding matrixes, wherein one of the 6 types ofprecoding matrix is configured in a form of $\begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1 \\X & 0 & 0\end{bmatrix},$ and satisfying a condition of${X \in \left\{ {1,\frac{1 + j}{\sqrt{2}},j,\frac{1 - j}{\sqrt{2}},{- 1},\frac{{- 1} - j}{\sqrt{2}},{- j},\frac{{- 1} + j}{\sqrt{2}}} \right\}},$and wherein individual rows of the precoding matrix respectivelycorrespond to four antennas of the 4 antennas, and individual columns ofthe precoding matrix respectively correspond to layers.